Definite Integral - Property of Definite Integral Property 1
- The definite integral is the limit of a Riemann sum.
- It represents the area under the curve between two points.
- The integral of a function can be thought of as the antiderivative of the function.
- The definite integral of a function over an interval is equal to the signed area between the curve and the x-axis.
- It provides a way to calculate the total accumulated change in a quantity over a given interval.
Definite Integral - Property of Definite Integral Property 2
- The definite integral is linear.
- The integral of the sum of two functions is equal to the sum of the integrals of each function.
- The integral of a constant times a function is equal to the constant times the integral of the function.
- This property allows us to simplify complex integrals by breaking them down into simpler integrals.
Definite Integral - Property of Definite Integral Property 3
- The definite integral is additive over intervals.
- Integrating a function over the union of two disjoint intervals is equal to the sum of integrating the function over each interval separately.
- The integral of a function from a to b plus the integral of the same function from b to c is equal to the integral of the function from a to c.
Definite Integral - Property of Definite Integral Property 4
- The definite integral can be used to calculate the average value of a function.
- The average value of a function is equal to the integral of the function over a given interval divided by the length of the interval.
- This property allows us to find the average value of a function over a specific range.
Definite Integral - Property of Definite Integral Property 5
- The definite integral can be used to calculate the total change in a quantity over a given interval.
- The integral of a rate of change function represents the total change in the quantity over the interval.
- This property is useful in calculating total displacement, total distance traveled, or total accumulated growth in a given interval.
Definite Integral - Property of Definite Integral Property 6
- The definite integral can be used to calculate the net signed area between a function and the x-axis.
- The integral of a non-negative function over an interval represents the area above the x-axis.
- The integral of a non-positive function over an interval represents the area below the x-axis.
- This property allows us to determine the net signed area between a curve and the x-axis.
Definite Integral - Property of Definite Integral Property 7
- The definite integral can be used to calculate the area between two curves.
- By subtracting the integral of one function from the integral of another function, we can find the area between the two curves.
- This property allows us to find the area enclosed by curves or the area of a region bounded by curves.
Definite Integral - Property of Definite Integral Property 8
- The definite integral can be used to calculate the average height of a function.
- By dividing the integral of a function by the length of the interval, we can find the average height of the function over the interval.
- This property is useful in finding the average value or average height of a function over a specific interval.
Definite Integral - Property of Definite Integral Property 9
- The definite integral can be used to calculate the volume of a solid of revolution.
- By rotating the area between a curve and the x-axis around the x-axis or y-axis, we create a three-dimensional shape.
- The volume of this shape can be found by integrating a function that represents the cross-sectional area of the shape.
- This property is useful in finding the volume of objects such as spheres, cylinders, and cones.
Definite Integral - Property of Definite Integral Property 10
- The definite integral can be used to calculate the work done by a variable force.
- By integrating the product of a force function and its displacement function, we can find the total work done.
- This property is useful in physics and engineering applications where work is done in moving objects or systems.
Slide 11: Property of Definite Integral Property 1
- The definite integral is the limit of a Riemann sum.
- It represents the area under the curve between two points.
- The integral of a function can be thought of as the antiderivative of the function.
- The definite integral of a function over an interval is equal to the signed area between the curve and the x-axis.
- It provides a way to calculate the total accumulated change in a quantity over a given interval.
Slide 12: Property of Definite Integral Property 2
- The definite integral is linear.
- The integral of the sum of two functions is equal to the sum of the integrals of each function.
- The integral of a constant times a function is equal to the constant times the integral of the function.
- This property allows us to simplify complex integrals by breaking them down into simpler integrals.
Slide 13: Property of Definite Integral Property 3
- The definite integral is additive over intervals.
- Integrating a function over the union of two disjoint intervals is equal to the sum of integrating the function over each interval separately.
- The integral of a function from a to b plus the integral of the same function from b to c is equal to the integral of the function from a to c.
Slide 14: Property of Definite Integral Property 4
- The definite integral can be used to calculate the average value of a function.
- The average value of a function is equal to the integral of the function over a given interval divided by the length of the interval.
- This property allows us to find the average value of a function over a specific range.
Slide 15: Property of Definite Integral Property 5
- The definite integral can be used to calculate the total change in a quantity over a given interval.
- The integral of a rate of change function represents the total change in the quantity over the interval.
- This property is useful in calculating total displacement, total distance traveled, or total accumulated growth in a given interval.
Slide 16: Property of Definite Integral Property 6
- The definite integral can be used to calculate the net signed area between a function and the x-axis.
- The integral of a non-negative function over an interval represents the area above the x-axis.
- The integral of a non-positive function over an interval represents the area below the x-axis.
- This property allows us to determine the net signed area between a curve and the x-axis.
Slide 17: Property of Definite Integral Property 7
- The definite integral can be used to calculate the area between two curves.
- By subtracting the integral of one function from the integral of another function, we can find the area between the two curves.
- This property allows us to find the area enclosed by curves or the area of a region bounded by curves.
Slide 18: Property of Definite Integral Property 8
- The definite integral can be used to calculate the average height of a function.
- By dividing the integral of a function by the length of the interval, we can find the average height of the function over the interval.
- This property is useful in finding the average value or average height of a function over a specific interval.
Definite Integral - Property of Definite Integral Property 1
- The definite integral is the limit of a Riemann sum.
- It represents the area under the curve between two points.
- The integral of a function can be thought of as the antiderivative of the function.
- The definite integral of a function over an interval is equal to the signed area between the curve and the x-axis.
- It provides a way to calculate the total accumulated change in a quantity over a given interval.
Definite Integral - Property of Definite Integral Property 2
- The definite integral is linear.
- The integral of the sum of two functions is equal to the sum of the integrals of each function.
- The integral of a constant times a function is equal to the constant times the integral of the function.
- This property allows us to simplify complex integrals by breaking them down into simpler integrals.
Definite Integral - Property of Definite Integral Property 3
- The definite integral is additive over intervals.
- Integrating a function over the union of two disjoint intervals is equal to the sum of integrating the function over each interval separately.
- The integral of a function from a to b plus the integral of the same function from b to c is equal to the integral of the function from a to c.
Definite Integral - Property of Definite Integral Property 4
- The definite integral can be used to calculate the average value of a function.
- The average value of a function is equal to the integral of the function over a given interval divided by the length of the interval.
- This property allows us to find the average value of a function over a specific range.
Definite Integral - Property of Definite Integral Property 5
- The definite integral can be used to calculate the total change in a quantity over a given interval.
- The integral of a rate of change function represents the total change in the quantity over the interval.
- This property is useful in calculating total displacement, total distance traveled, or total accumulated growth in a given interval.
Definite Integral - Property of Definite Integral Property 6
- The definite integral can be used to calculate the net signed area between a function and the x-axis.
- The integral of a non-negative function over an interval represents the area above the x-axis.
- The integral of a non-positive function over an interval represents the area below the x-axis.
- This property allows us to determine the net signed area between a curve and the x-axis.
Definite Integral - Property of Definite Integral Property 7
- The definite integral can be used to calculate the area between two curves.
- By subtracting the integral of one function from the integral of another function, we can find the area between the two curves.
- This property allows us to find the area enclosed by curves or the area of a region bounded by curves.
Definite Integral - Property of Definite Integral Property 8
- The definite integral can be used to calculate the average height of a function.
- By dividing the integral of a function by the length of the interval, we can find the average height of the function over the interval.
- This property is useful in finding the average value or average height of a function over a specific interval.
Definite Integral - Example
- Let’s consider the function f(x) = x^2.
- We want to find the area under the curve between x = 0 and x = 2.
- Using the definite integral, we can calculate this by evaluating ∫[0, 2] x^2 dx.
- The antiderivative of x^2 is (1/3)x^3.
- Evaluating the definite integral gives us [(1/3) * (2^3)] - [(1/3) * (0^3)] = (8/3) - 0 = 8/3.
- Therefore, the area under the curve f(x) = x^2 between x = 0 and x = 2 is 8/3.
Definite Integral - Property of Definite Integral Property 9
- The definite integral can be used to calculate the volume of a solid of revolution.
- By rotating the area between a curve and the x-axis around the x-axis or y-axis, we create a three-dimensional shape.
- The volume of this shape can be found by integrating a function that represents the cross-sectional area of the shape.
- This property is useful in finding the volume of objects such as spheres, cylinders, and cones.
Definite Integral - Property of Definite Integral Property 10
- The definite integral can be used to calculate the work done by a variable force.
- By integrating the product of a force function and its displacement function, we can find the total work done.
- This property is useful in physics and engineering applications where work is done in moving objects or systems.
Definite Integral - Example
- Let’s consider a variable force F(x) = 2x acting on an object that moves along the x-axis from x = 0 to x = 3.
- To find the work done by this force, we integrate the product of the force and the displacement function.
- The displacement function is given by Δx = x - 0 = x.
- The work done is given by W = ∫[0, 3] F(x) * Δx.
- Substituting the force and displacement functions, we have W = ∫[0, 3] 2x * x dx.
- Evaluating the definite integral gives us [(2/3) * (3^3)] - [(2/3) * (0^3)] = 18.
- Therefore, the work done by the force F(x) = 2x over the interval x = 0 to x = 3 is 18 units.
Definite Integral - Recap
- The definite integral has several properties that make it a powerful tool in calculus.
- It can be used to calculate the area under a curve, the average value of a function, and the total change in a quantity.
- It is linear, additive over intervals, and can be used to calculate the net signed area between a curve and the x-axis.
- The definite integral can also be used to find the volume of solids of revolution and the work done by variable forces.