Calculus Help intergrals

Question

Evaluate the integral by interpreting it in terms of areas.

0		5 +	1 − x2		dx
−1

Answer 1

∫_{(from -1 to 0)} [5 + √(1-x²)] dx = area of the region above the x-axis between x = -1 and x = 0 and below the graph of the integrand (because the integrand is nonnegative between the limits of integration).

Between x = -1 and x = 0, the graph of y = √(1 - x²) is 1/2 of the upper half of the circle centered at (0,0) with radius 1. Adding 5 translates the graph upward 5 units.

If you draw a picture, you will see that the region consists of a 1 by 5 rectangle and 1/4 of a circle with radius 1.

So, the value of the integral is (1)(5) + (1/4)π(1)² = 5 + π/4.

Answer 2

We have ∫_-1⁰(5 + sqrt(1 - x²)dx. Now, y = 5 + sqrt(1 - x²), or (y - 5)² + x² = 1. This is circle with center at (0,5) and radius r = 1. Because this integral is equal area of upper quater of this cercle for x from [-1,0] and equal π/4.

Answer: This integral equal π/4.