# 7.E: Integration (Exercises)

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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 7.1.1**Suppose 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 7.1.2**Suppose 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 7.1.3**By 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 7.1.4**By 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 7.1.5**By 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 7.1.6**By 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 7.1.7**Let $\ds f(x)=x^2+3x+2$. Approximate the area under the curve between $x=0$ and $x=2$ using 4 rectangles and also using 8 rectangles. (answer)

**Ex 7.1.8**Let $\ds f(x)=x^2-2x+3$. 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 7.2.17**Find the derivative of \( G(x)=\int_1^x t^2-3t\,dt\) (answer)

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

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

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

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

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

## 7.3: Some Properties of Integrals

**Ex 7.3.1**An 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 7.3.2**An 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 7.3.3**An 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 7.3.4**Consider 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 7.3.5**Consider 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 7.3.6**Evaluate 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)

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