SATHEE: Area Question 36

Area Question 36

Question 36

If $f (x)$ is a continuous function given by

$f (x) = \begin{array}{ll} 2 x, & | x | \leq 1 x^{2} + a x + b, & | x | > 1 \end{array}$

Then, find the area of the region in the third quadrant bounded by the curves $x = - 2 y^{2}$ and $y = f (x)$ lying on the left on the line $8 x + 1 = 0$ .

$(1999, 5$ M)

Show Answer

Solution:

Given, $f (x) = \begin{aligned} 2 x, & x^{2} + a x + b, | x | > 1 \end{aligned}$

$\Rightarrow f (x) = \begin{array}{ll} x^{2} + a x + b, & if x < - 1 2 x, & if - 1 \leq x < 1 x^{2} + a x + b, & if x \geq 1 \end{array}$

$f$ is continuous on $R$ , so $f$ is continuous at -1 and 1 .

$\begin{aligned} lim_{x \to - 1^{-}} f (x) = lim_{x \to - 1^{+}} f (x) = f (- 1) & and lim_{x \to 1^{-}} f (x) = lim_{x \to 1^{+}} f (x) = f (1) & \Rightarrow 1 - a + b = - 2 and 2 = 1 + a + b & \Rightarrow a - b = 3 and a + b = 1 & ∴ a = 2, b = - 1 & Hence, f (x) = \begin{array}{llc} x^{2} + 2 x - 1, & if & x < - 1 2 x, & if & - 1 \leq x < 1 x^{2} + 2 x - 1, & if & x \geq 1 \end{array} \end{aligned}$

Next, we have to find the points $x = - 2 y^{2}$ and $y = f (x)$ . The point of intersection is $(- 2, - 1)$ .

$∴$ Required area $= \int_{- 2}^{- 1 / 8} \sqrt{\frac{- x}{2}} - f (x) d x$

$$ \begin{aligned} & =\int_{-2}^{-1 / 8} \sqrt{\frac{-x}{2}} d x-\int_{-2}^{-1}\left(x^{2}+2 x-1\right) d x-\int_{-1}^{-1 / 8} 2 x d x \ & =-\frac{2}{3 \sqrt{2}}\left[(-x)^{3 / 2}\right]{-2}^{-1 / 8}-\frac{x^{3}}{3}+x^{2}-x-{-2}^{-\left[x^{2}\right]_{-1}^{-1 / 8}} \ & =-\frac{2}{3 \sqrt{2}} \frac{1^{3 / 2}}{8}-2^{3 / 2}–\frac{1}{3}+1+1 \ & +-\frac{8}{3}+4+2-\frac{1}{64}-1 \ & =\frac{\sqrt{2}}{3}\left[2 \sqrt{2}-2^{-9 / 2}\right]+\frac{5}{3}+\frac{63}{64} \ & =\frac{63}{16 \times 3}+\frac{509}{64 \times 3}=\frac{761}{192} \text { squnits } \end{aligned} $$

Area Question 36

Question 36

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Area Question 36

Question 36

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu