Graph the integrand and use known area formulas to evaluate the integrals.

Question

Graph the integrand and use known area formulas to evaluate the integrals.

∫-1 to 1 (1+√(1-x²))dx

Answer 1

If y = √(1 - x²), then y² = 1 - x². So, x²+ y² = 1 (the circle centered at (0,0) with radius 1).

The graph of y = √(1-x²) is the upper half of the circle.

Translate 1 unit upward to obtain the graph of y = 1 + √(1-x²).

Between x = -1 and x = 1, the region is bounded above by the semicircle and below by the x-axis.

Divide the region into a semicircle of radius 1 and a rectangle (with length 2 and height 1).

∫_{(-1 to 1)} [1 + √(1-x²)]dx = (Area of semicircle) + (Area of rectangle) = π/2 + 2