Find the area of the region bounded by the graphs of the algebraic functions.

Question

Quizplus · Accepted Answer

To find the area bounded by the two curves, we need to find the points of intersection first. Setting the two equations equal to each other, we get:
$$25-x^2=x+4$$
$$x^2+x-29=0$$
Solving for $x$ gives $x=4$ (rejecting the negative solution).

Thus the area of the region bounded by the two curves is:

$$\int_{-4}^{4} (25-x^2)-(x+4) dx=\int_{-4}^{4} 21-x-x^2 dx$$

Using the power rule, the integral evaluates to:

$$\left[ 21x - \frac{x^2}{2} - \frac{x^3}{3} \right]_{-4}^{4}= \frac{512}{3}$$

So the answer is C.

Find the Area of the Region Bounded by the Graphs