\( \int _ { 0 } ^ { 1 } x ^ { 2 } - \sqrt { 1 - ( x - 1 ) ^ { 2 } } d x = \)

To solve the integral \( \int_{0}^{1} \left( x^2 - \sqrt{1 - (x - 1)^2} \right) dx \), we first simplify the expression under the integral sign. The term \( \sqrt{1 - (x - 1)^2} \) can be rewritten as \( \sqrt{1 - (x^2 - 2x + 1)} = \sqrt{2x - x^2} \). Thus, the integral becomes: \[ \int_{0}^{1} \left( x^2 - \sqrt{2x - x^2} \right) dx \] Next, let's calculate the individual components of the integral: 1. First compute \( \int_{0}^{1} x^2 dx \): \[ \int_{0}^{1} x^2 dx = \left[\frac{x^3}{3}\right]_{0}^{1} = \frac{1}{3} \] 2. Now compute \( \int_{0}^{1} \sqrt{2x - x^2} dx \). To do this, we can simplify \( 2x - x^2 \) further; it can be factored as \( x(2-x) \), which resembles a semi-circle when integrated. Using substitution, let's set \( x = 2 \sin^2(t) \), then \( dx = 4 \sin(t) \cos(t) dt \). The limits change from \( 0 \) to \( \frac{\pi}{4} \) as \( x \) varies from \( 0 \) to \( 1 \). With this substitution, the expression for \( \sqrt{2x - x^2} \) becomes: \[ \sqrt{2(2 \sin^2(t)) - (2 \sin^2(t))^2} = \sqrt{4 \sin^2(t) - 4 \sin^4(t)} = 2 \sin(t) \sqrt{1 - \sin^2(t)} = 2 \sin(t) \cos(t) \] Thus, \[ \int_{0}^{1} \sqrt{2x - x^2} dx = \int_{0}^{\frac{\pi}{4}} 2 \sin(t) \cos(t) (4 \sin(t) \cos(t) dt) = \int_{0}^{\frac{\pi}{4}} 8 \sin^2(t) \cos^2(t) dt \] This can be simplified using the identity \( \sin^2(t) \cos^2(t) = \frac{1}{4} \sin^2(2t) \): \[ 8 \int_{0}^{\frac{\pi}{4}} \frac{1}{4} \sin^2(2t) dt = 2 \int_{0}^{\frac{\pi}{4}} \sin^2(2t) dt \] Recognizing that \( \int \sin^2(u) du = \frac{u}{2} - \frac{\sin(2u)}{4} \), we can evaluate the integral from \( 0 \) to \( \frac{\pi}{2} \) for \( 2t \) yielding: \[ = 2 \left[ \frac{2t}{4} - \frac{\sin(4t)}{8} \right]_{0}^{\frac{\pi}{4}}= 2\left( \frac{\pi/4}{2} - 0 \right) = \frac{\pi}{4} \] Combining both components, we have \[ \int_{0}^{1} \left(x^2 - \sqrt{2x - x^2}\right) dx = \frac{1}{3} - \frac{\pi}{4} \] Therefore, the final answer is: \[ \int _ { 0 } ^ { 1 } x ^ { 2 } - \sqrt { 1 - ( x - 1 ) ^ { 2 } } d x = \frac{1}{3} - \frac{\pi}{4} \]