4.5E: Exercises

Exercise $\PageIndex{1}$

1. How are the definite and indefinite integrals related?

2. What constants of integration is most commonly used when evaluating definite integrals?

3. T/F: If $f$ is a continuous function, then $F(x) =\int_a^x f(t)\,dt$ is also a continuous function.

4. The definite integral can be used to find "the area under a curve." Give two other uses for definite integrals.

Answer

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Exercise $\PageIndex{2}$

Evaluate the definite integrals

1. $\int_1^3 (3x^2-2x+1)\,dx$

2. $\int_0^4 (x-1)^2\,dx$

3. $\int_{-1}^1 (x^3-x^5)\,dx$

4. $\int_{\pi/2}^{\pi}\cos x\,dx$

5. $\int_0^{\pi/4}\sec^2 x\,dx$

6. $\int_1^e \frac{1}{x}\,dx$

7. $\int_{-1}^1 5^x \,dx$

8. $\int_{-2}^{-1}(4-2x^3)\,dx$

9. $\int_0^{\pi}(2\cos x -2\sin x)\,dx$

10. $\int_1^3 e^x\,dx$

11. $\int_0^4 \sqrt{t}\,dt$

12. $\int_9^{25} \frac{1}{\sqrt{t}}\,dt$

13. $\int_1^8 \sqrt[3]{x}\,dx$

14. $\int_1^2 \frac{1}{x}\,dx$

15. $\int_1^2 \frac{1}{x^2}\,dx$

16. $\int_1^2 \frac{1}{x^3}\,dx$

17. $\int_0^1 x\,dx$

18. $\int_0^1 x^2\,dx$

19. $\int_0^1 x^3\,dx$

20. $\int_0^1 x^{100}\,dx$

21. $\int_{-4}^4 dx$

22. $\int_{-10}^{-5} 3\,dx$

23. $\int_{-2}^2 0\,dx$

24. $\int_{\pi/6}^{\pi/3}\csc x \cot x\,dx$

Answer

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Exercise $\PageIndex{3}$

29. Explain why:
(a) $\int_{-1}^1 x^n\,dx=0$, when n is a positive, odd integer, and
(b) $\int_{-1}^1x^n\,dx =2\int_0^1 x^n \,dx$ when n is a positive, even integer.

Answer

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Exercise $\PageIndex{4}$

Find a value c guaranteed by the Mean Value Theorem.

1. $\int_0^2 x^2\,dx$

2. $\int_{-2}^2 x^2\,dx$

3. $\int_0^1 e^x\,dx$

4. $\int_0^16 \sqrt{x}\,dx$

Answer

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Exercise $\PageIndex{5}$

Find the average value of the function on the given interval.

1. $f(x) =\sin x \text{ on }[0,\pi/2]$

2. $y =\sin x \text{ on }[0,\pi]$

3. $y = x \text{ on }[0,4]$

4. $y =x^2 \text{ on }[0,4]$

5. $y =x^3 \text{ on }[0,4]$

6. $g(t) =1/t \text{ on }[1,e]$

Answer

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Exercise $\PageIndex{6}$

A velocity function of an object moving along a straight line is given. Find the displacement of the object over the given time interval.

1. $v(t) =-32t+20$ft/s on [0,5].

2. $v(t) =-32t+200$ft/s on [0,10].

3. $v(t) =2^t$mph on [-1,1].

4. $v(t) =\cos t$ft/s on $[0,3\pi /2]$.

5. $v(t) =\sqrt[4]{t}$ft/s on [0,16].

Answer

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Exercise $\PageIndex{7}$

An acceleration function of an object moving along a straight line is given. Find the change of the object's velocity over the given time interval.

1. $a(t) =-32$ft/s on [0,2].

2. $a(t) =10$ft/s on [0,5].

3. $a(t) =t$ft/s$^2$ on [0,2].

4. $a(t) =\cos t$ft/s$^2$ on $[0,\pi]$.

Answer

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Exercise $\PageIndex{8}$

Sketch the given functions and find the area of the enclosed region.

1. $y =2x,\, y=5x,\text{ and }x=3$.

2. $y=-x+1,\,y=3x+6,\,x=2\text{ and }x=-1$.

3. $y=x^2-2x+5,\,y=5x-5$.

4. $y = 2x^2+2x-5,\,y=x^2+3x+7$.

Answer

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Exercise $\PageIndex{9}$

Find $F'(x)$.

1. $F(x) =\int_2^{x^3+x}\frac{1}{t}\,dt$

2. $F(x) = \int_{x^3}^0 t^3\,dt$

3. $F(x)=\frac{x}{x^2}(t+2)\,dt$

4. $F(x) =\int_{\ln x}^{e^x}\sin t\,dt$

Answer

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