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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