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Evaluating a geometric integral (semicircle plus rectangle)

Consider the function f(x)=1−x2+1f(x) = \sqrt{1-x^2} + 1 on the interval [−1,1][-1,1].

Use the fact that y=1−x2y = \sqrt{1-x^2} describes the upper half of the unit circle centered at the origin to evaluate

∫−11(1−x2+1) dx\int_{-1}^{1} \left(\sqrt{1-x^2} + 1\right)\,dx

without computing an antiderivative. Sketch or picture the region under f(x)f(x) on [−1,1][-1,1], identify it as a combination of simple shapes, and use area formulas to find the exact value of the integral.

Evaluating a geometric integral (semicircle plus rectangle) graph
Hint

Split the region into two pieces: the part under y=1y=1 (a rectangle) and the part between y=1y=1 and y=f(x)y=f(x) (a semicircle of radius 1). Use linearity of the integral to add their areas.

Step-by-step solution

Split f(x)=1−x2+1f(x) = \sqrt{1-x^2} + 1 using linearity of the integral: ∫−111−x2 dx+∫−111 dx\int_{-1}^{1} \sqrt{1-x^2}\,dx + \int_{-1}^{1} 1\,dx. The first integral represents the area of the upper half of the unit circle (radius 11), which is a semicircle.

The semicircle has area 12π(1)2=π2\frac{1}{2}\pi (1)^2 = \frac{\pi}{2}. The second integral is the area of a rectangle with width 22 (from −1-1 to 11) and height 11, giving area 22. Adding these gives the total: π2+2\frac{\pi}{2} + 2.

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