Question

Evaluate the integral below by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using geometry.-5 ∫5 sqrt(5^2−x^2) dx my lower limit is -5 and my upper is 5. How do I solve this problem? do i graph this equation and use left or right approximation?

Answer 1

`(25\pi)/(2)\approx39.27\text{ units}^2`

Explanation:

I assume your problem is
`\displaystyle \int\limits^(5)_(-5) {√(5^2-x^2)} \, dx`. Recognize that
`√(5^2-x^2)=√(25-x^2)` is a semi-circle directly above the x-axis with a radius of 5. The bounds cover the whole area of the semicircle, so its area is just half the area of a circle with a radius of 5. You probably know that the area of a circle is
`A=\pi r^2`, so the area of a semicircle would be
`A=(\pi r^2)/(2)`. Thus, the area using geometry is
`A=(\pi (5)^2)/(2)=(25\pi)/(2)\approx39.27\text{ units}^2`.