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# 4.5E: Exercises

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## 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.

Under Construction

## 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. $$\int_1^8 \sqrt{x}\,dx$$

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

Under Construction

## 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.

Under Construction

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

Under Construction

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

Under Construction

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

Under Construction

## 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 $$[0,\pi]$$.

Under Construction

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

Under Construction

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