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Answer 1 · 2017-06-18T06:49:17

2 $int_0^3$ [ $color(blue)(15 - x^2)$ - $color(red)(x^2 + 3)$ ] dx = 96

Find the area between the two curves. The area can be found at the points of intersection.

Find the points of intersection.

15 - $x^2$ = $x^2$ - 3
15 + 3 = $x^2$ + $x^2$
18 = 2 $x^2$
$x^2$ = 9
x = 3

enter image source here

From the graph, we can see that our equation is this.

$int_-3^3$ [ $color(blue)(15 - x^2)$ - ( $color(red)(x^2 - 3)$ )] dx

Since the graph is symmetrical, we start our function from this equation.

2 $int_a^b$ [ $color(blue)(f(x))$ - $color(red)(g(x))$ ] dx

Now our function looks like this.

2 $int_0^3$ [ $color(blue)(15 - x^2)$ - ( $color(red)(x^2 - 3)$ )] dx

We evaluate.

2 $int_0^3$ [ $color(blue)(15 - x^2)$ - $color(red)(x^2 + 3)$ ] dx

2 $int_0^3$ $[- 2x^2 + 18]$ dx

Take the antiderivative and continue to evaluate.

2 $[-2/3 x^3 + 18x]_0^3$

2 $[-2/3 (3)^3 + 18(3)] - [-2/3 0^3 + 18(0)]$

2 $[-2/3 (9) + 54] - [0 + 0]$

2 $[-18/3 + 54]$

2 [-6 + 54]

2 [48]

96