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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