Find the value of c for which the area enclosed by the curves y=c-x^2 and y=x^2-c is equal to 24.

Answer 1

To find the bounds of integration, we need to find the x-coordinates of the points of intersection:

c - x² = x² - c

x = ± √c

Since c - x² is above, the area between the curves is given by ∫_-√c^√c (c - x²) - (x² - c) dx

= 2 ∫_-√c^√c (c - x²) dx = 24

cx - 1/3x³ ]^√c_-√c = 12

c√c - 1/3c√c - (-c√c + 1/3c√c) = 12

4/3c√c = 12

c√c = 9

c = ³√81 = 3 ³√3

Answer 2

Hi,

y₁=c-x^2 and y₂=x^2-c

First Of all, find intersection between two curves

c-x^2 =x^2-c =====> 2x² = 2c =====> x = ±√c

area between two curve is ∫(2c-2x²)dx = ∫2(c-x²)dx = 2∫(c-x²)dx when it bounded between -√c to √c or choose 0 to √c times 2

( the same result )

You know the answer of integral is 4[cx - 1/3 x³] now calculate between 0 to √c

4[cx - 1/3 x³] = 4[c√c -1/3 (√c)³] = 4 [ c√c -1/3 c√c] = 4[ 2/3c√c] = 24 after simplify you will have

c√c = 9 ======> c = 3 (3)^1/3

I hope it is useful,

Minoo