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7.E: Integration (Exercises)

Last updated

Nov 10, 2020
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- 7.3: Some Properties of Integrals
- 8: Techniques of Integration

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These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.

7.1: Two Examples

Ex 7.1.1Suppose an object moves in a straight line so that its speed at time $t$ is given by $v(t)=2t+2$, and that at $t=1$ the object is at position 5. Find the position of the object at $t=2$. (answer)

Ex 7.1.2Suppose an object moves in a straight line so that its speed at time $t$ is given by $\ds v(t)=t^2+2$, and that at $t=0$ the object is at position 5. Find the position of the object at $t=2$. (answer)

Ex 7.1.3By a method similar to that in example 7.1.2, find the area under $y=2x$ between $x=0$ and any positive value for $x$. (answer)

Ex 7.1.4By a method similar to that in example 7.1.2, find the area under $y=4x$ between $x=0$ and any positive value for $x$. (answer)

Ex 7.1.5By a method similar to that in example 7.1.2, find the area under $y=4x$ between $x=2$ and any positive value for $x$ bigger than 2. (answer)

Ex 7.1.6By a method similar to that in example 7.1.2, find the area under $y=4x$ between any two positive values for $x$, say $a < b$. (answer)

Ex 7.1.7Let $\ds f(x)=x^2+3x+2$. Approximate the area under the curve between $x=0$ and $x=2$ using 4 rectangles and also using 8 rectangles. (answer)

Ex 7.1.8Let $\ds f(x)=x^2-2x+3$. Approximate the area under the curve between $x=1$ and $x=3$ using 4 rectangles. (answer)

7.2: The Fundamental Theorem of Calculus

Find the antiderivatives of the functions:

Ex 7.2.1 $8\sqrt{x}$ (answer)

Ex 7.2.2 $3t^2+1$ (answer)

Ex 7.2.3 $4/\sqrt{x}$ (answer)

Ex 7.2.4 $2/z^2$ (answer)

Ex 7.2.5 $7s^{-1}$ (answer)

Ex 7.2.6 $(5x+1)^2$ (answer)

Ex 7.2.7 $(x-6)^2$ (answer)

Ex 7.2.8 $x^{3/2}$ (answer)

Ex 7.2.9 ${2\over x\sqrt x}$ (answer)

Ex 7.2.10 $|2t-4|$ (answer)

Compute the values of the integrals:

Ex 7.2.11 $\int_1^4 t^2+3t\,dt$ (answer)

Ex 7.2.12 $\int_0^\pi \sin t\,dt$ (answer)

Ex 7.2.13 $\int_1^{10} {1\over x}\,dx$ (answer)

Ex 7.2.14 $\int_0^5 e^x\,dx$ (answer)

Ex 7.2.15 $\int_0^3 x^3\,dx$ (answer)

Ex 7.2.16 $\int_1^2 x^5\,dx$ (answer)

Ex 7.2.17Find the derivative of $G(x)=\int_1^x t^2-3t\,dt$ (answer)

Ex 7.2.18Find the derivative of $G(x)=\int_1^{x^2} t^2-3t\,dt$ (answer)

Ex 7.2.19Find the derivative of $G(x)=\int_1^x e^{t^2}\,dt$ (answer)

Ex 7.2.20Find the derivative of $G(x)=\int_1^{x^2} e^{t^2}\,dt$ (answer)

Ex 7.2.21Find the derivative of $G(x)=\int_1^x \tan(t^2)\,dt$ (answer)

Ex 7.2.22Find the derivative of $G(x)=\int_1^{x^2} \tan(t^2)\,dt$ (answer)

7.3: Some Properties of Integrals

Ex 7.3.1An object moves so that its velocity at time $t$ is $v(t)=-9.8t+20$ m/s. Describe the motion of the object between $t=0$ and $t=5$, find the total distance traveled by the object during that time, and find the net distance traveled. (answer)

Ex 7.3.2An object moves so that its velocity at time $t$ is $v(t)=\sin t$. Set up and evaluate a single definite integral to compute the net distance traveled between $t=0$ and $t=2\pi$. (answer)

Ex 7.3.3An object moves so that its velocity at time $t$ is $v(t)=1+2\sin t$ m/s. Find the net distance traveled by the object between $t=0$ and $t=2\pi$, and find the total distance traveled during the same period. (answer)

Ex 7.3.4Consider the function $f(x)=(x+2)(x+1)(x-1)(x-2)$ on $[-2,2]$. Find the total area between the curve and the $x$-axis (measuring all area as positive). (answer)

Ex 7.3.5Consider the function $\ds f(x)=x^2-3x+2$ on $[0,4]$. Find the total area between the curve and the $x$-axis (measuring all area as positive). (answer)

Ex 7.3.6Evaluate the three integrals:

$A=\int_0^3 (-x^2+9)\,dx\qquad B=\int_0^{4} (-x^2+9)\,dx\qquad C=\int_{4}^3 (-x^2+9)\,dx,$ and verify that $A=B+C$. (answer)

Contributors

David Guichard (Whitman College)

7.1: Two Examples

7.2: The Fundamental Theorem of Calculus

Find the antiderivatives of the functions:

7.3: Some Properties of Integrals

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