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Section 4.3

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1. (a) \displaystyle{\int_0^1 xdx = \frac{1}{3}}

(c) \displaystyle{\int_0^2 x^3dx = 4}

(e) \displaystyle{\int_0^2 (2x^3 + 3x^2 +x - 4)dx = 10}

(g) \displaystyle{\int_1^4 \frac{1}{\sqrt{t}} \ dt = 2}

(i) \displaystyle{\int_0^{\frac{\pi}{2}} \sin(x)dx = 1}

(j) \displaystyle{\int_{-\pi}^\pi \cos(z)dz = 0}

2. (a) \displaystyle{\int_0^2 (x + 1)^2 dx = \frac{26}{3}}

(c) \displaystyle{\int_0^4 \sqrt{1 + 2t} \ dt = \frac{26}{3}}

(e) \displaystyle{\int_0^\pi \sin(2x)dx = 0}

(g) \displaystyle{\int_0^{\frac{\pi}{3}} 4\sin(3x)dx = \frac{8}{3}}

(i) \displaystyle{\int_0^{\sqrt{\pi}} 2x\sin(x^2)dx = 2}

(k) \displaystyle{\int_0^{\sqrt{\pi}} x\sin(x^2)dx = 1}

4. (a) F'(x) = \sin^2(4x)

(c) F'(x) = -\cos^3(x)

(e) \displaystyle{f'(x) = \frac{2x}{1 + x^4}}

(f) \displaystyle{h'(z) = 3\sqrt{1 + 9z^2} - \sqrt{1 + z^2}}

5. (a) \displaystyle{\frac{d}{dx}\int_1^x \frac{1}{1 + t^2} \ dt = \frac{1}{1 + x^2}}

(c) \displaystyle{\frac{d}{dt}\int_{t^2}^5 \sin^2(3x)dx = -2t\sin^2(3t^2)}

6. Area \displaystyle{= \int_0^{\frac{\pi}{2}} 3\sin(2x)dx = 3}

7. Area \displaystyle{= \int_0^1 x^2dx + \int_1^2 (x - 2)^2dx = \frac{2}{3}}

8. Area \displaystyle{= \int_0^1 (x - x^2)dx = \frac{1}{6}}

9. Area \displaystyle{= \int_{-1}^1 (2 - 2x^2)dx = \frac{8}{3}}

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