?Displacement from velocity A particle moves along a line with a velocity given by

Problem 1P A 64 kg woman stands on a very light, rigid board that rests on a bathroom scale at each end, as shown in Figure 1. What is the reading on each of the scales? FIGURE 1

Problem 1CQ A 64 kg student stands on a very light, rigid board that rests on a bathroom scale at each end, as shown in Figure P8.1 . What is the reading on each of the scales?

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume f and \(f^{\prime}\) are continuous functions for all real numbers. a. If \(A(x)=\int_{a}^{x} f(t) d t\) and f(t )= 2t - 3, then A is a quadratic function. b. Given an area function \(A(x)=\int_{a}^{x} f(t) d t\) and an antiderivative F of f, it follows that \(A^{\prime}(x)=F(x)\). c. \(\int_{a}^{b} f^{\prime}(x) d x=f(b)-f(a)\) d. If \(\int_{a}^{b}|f(x)| d x=0\), then f(x) = 0 on [a, b]. e. If the average value of f on [a, b] is zero, then f(x) = 0 on [a, b]. f. \(\int_{a}^{b}(2 f(x)-3 g(x)) d x=2 \int_{a}^{b} f(x) d x+3 \int_{b}^{a} g(x) d x\) g. \(\int f^{\prime}(g(x)) g^{\prime}(x) d x=f(g(x))+C\)

Velocity to displacement An object travels on the x-axis with a velocity given by v(t) = 2t + 5, for \(0 \leq t \leq 4\). a. How far does the object travel for \(0 \leq t \leq 4\)? b. What is the average value of v on the interval [0, 4] ? c. True or false: The object would travel as far as in part (a) if it traveled at its average velocity (a constant) for \(0 \leq t \leq 4\).

Area by geometry Use geometry to evaluate \(\int_{0}^{7} f(x) d x\), where the graph of f is given in the figure.

Displacement by geometry Use geometry to find the displacement of an object moving along a line for \(0 \leq t \leq 8\), where the graph of its velocity v = g(t) is given in the figure.

Area by geometry Use geometry to evaluate \(\int_{0}^{4} \sqrt{8 x-x^{2}} d x\) (Hint: Complete the square of \(8 x-x^{2}\) first).

Bagel output The manager of a bagel bakery collects the following production rate data (in bagels per minute) at six different times during the morning. Estimate the total number of bagels produced between 6:00 and 7:30 a.m.

Integration by Riemann sums Consider the integral \(\int_{1}^{4}(3 x-2) d x\) a. Give the right Riemann sum for the integral with n = 3. b. Use summation notation to write the right Riemann sum for an arbitrary positive integer n. c. Evaluate the definite integral by taking the limit as \(n \rightarrow \infty\) of the Riemann sum in part (b).

Evaluating Riemann sums Consider the function f(x) = 3x + 4 on the interval [3, 7]. Show that the midpoint Riemann sum with n = 4 gives the exact area of the region bounded by the graph.

Sum to integral Evaluate the following limit by identifying the integral that it represents: \(\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left[\left(\frac{4 k}{n}\right)^{8}+1\right]\left(\frac{4}{n}\right)\).

Area function by geometry Use geometry to find the area A(x) that is bounded by the graph of f(t) = 2t - 4 and the t-axis between the point (2, 0) and the variable point (x, 0), where \(x \geq 2\). Verify that \(A^{\prime}(x)=f(x)\).

Evaluating integrals Evaluate the following integrals. \(\int_{-2}^{2}\left(3 x^{4}-2 x+1\right) d x\)

Evaluating integrals Evaluate the following integrals. \(\int \cos 3 x d x\)

Evaluating integrals Evaluate the following integrals. \(\int_{0}^{2}(x+1)^{3} d x\)

Evaluating integrals Evaluate the following integrals. \(\int_{0}^{1}\left(4 x^{21}-2 x^{16}+1\right) d x\)

Evaluating integrals Evaluate the following integrals. \(\int\left(9 x^{8}-7 x^{6}\right) d x\)

Evaluating integrals Evaluate the following integrals. \(\int_{-2}^{2} e^{4 x+8} d x\)

Evaluating integrals Evaluate the following integrals. \(\int_{0}^{1} \sqrt{x}(\sqrt{x}+1) d x\)

Evaluating integrals Evaluate the following integrals. \(\int \frac{x^{2}}{x^{3}+27} d x\)

Evaluating integrals Evaluate the following integrals. \(\int_{0}^{1} \frac{d x}{\sqrt{4-x^{2}}}\)

Evaluating integrals Evaluate the following integrals. \(\int y^{2}\left(3 y^{3}+1\right)^{4} d y\)

Evaluating integrals Evaluate the following integrals. \(\int_{0}^{3} \frac{x}{\sqrt{25-x^{2}}} d x\)

Evaluating integrals Evaluate the following integrals. \(\int x \sin x^{2} \cos ^{8} x^{2} d x\)

Evaluating integrals Evaluate the following integrals. \(\int \sin ^{2} 5 \theta d \theta\)

Evaluating integrals Evaluate the following integrals. \(\int_{0}^{\pi}\left(1-\cos ^{2} 3 \theta\right) d \theta\)

Evaluating integrals Evaluate the following integrals. \(\int \frac{x^{2}+2 x-2}{x^{3}+3 x^{2}-6 x} d x\)

Evaluating integrals Evaluate the following integrals. \(\int_{0}^{\ln 2} \frac{e^{x}}{1+e^{2 x}} d x\)

Symmetry properties Suppose that \(\int_{0}^{4} f(x) d x=10\) and \(\int_{0}^{4} g(x) d x=20\). Furthermore, suppose that f is an even function and g is an odd function. Evaluate the following integrals. a. \(\int_{-4}^{4} f(x) d x\) b. \(\int_{-4}^{4} 3 g(x) d x\) c. \(\int_{-4}^{4}(4 f(x)-3 g(x)) d x\)

Properties of integrals The figure shows the areas of regions bounded by the graph of f and the x-axis. Evaluate the following integrals. a. \(\int_{a}^{c} f(x) d x\) b. \(\int_{b}^{d} f(x) d x\) c. \(2 \int_{c}^{b} f(x) d x\) d. \(4 \int_{a}^{d} f(x) d x\) e. \(3 \int_{a}^{b} f(x) d x\) f. \(2 \int_{b}^{d} f(x) d x\)

Properties of integrals Suppose that \(\int_{1}^{4} f(x) d x=6\), \(\int_{1}^{4} g(x) d x=4\), and \(\int_{3}^{4} f(x) d x=2\). Evaluate the following integrals or state that there is not enough information. \(\int_{1}^{4} 3 f(x) d x\)

Properties of integrals Suppose that \(\int_{1}^{4} f(x) d x=6\), \(\int_{1}^{4} g(x) d x=4\), and \(\int_{3}^{4} f(x) d x=2\). Evaluate the following integrals or state that there is not enough information. \(-\int_{4}^{1} 2 f(x) d x\)

Properties of integrals Suppose that \(\int_{1}^{4} f(x) d x=6\), \(\int_{1}^{4} g(x) d x=4\), and \(\int_{3}^{4} f(x) d x=2\). Evaluate the following integrals or state that there is not enough information. \(\int_{1}^{4}(3 f(x)-2 g(x)) d x\)

Properties of integrals Suppose that \(\int_{1}^{4} f(x) d x=6\), \(\int_{1}^{4} g(x) d x=4\), and \(\int_{3}^{4} f(x) d x=2\). Evaluate the following integrals or state that there is not enough information. \(\int_{1}^{4} f(x) g(x) d x\)

Properties of integrals Suppose that \(\int_{1}^{4} f(x) d x=6\), \(\int_{1}^{4} g(x) d x=4\), and \(\int_{3}^{4} f(x) d x=2\). Evaluate the following integrals or state that there is not enough information. \(\int_{1}^{3} \frac{f(x)}{g(x)} d x\)

Properties of integrals Suppose that \(\int_{1}^{4} f(x) d x=6\), \(\int_{1}^{4} g(x) d x=4\), and \(\int_{3}^{4} f(x) d x=2\). Evaluate the following integrals or state that there is not enough information. \(\int_{3}^{1}(f(x)-g(x)) d x\)

Displacement from velocity A particle moves along a line with a velocity given by \(v(t)=5 \sin (\pi t)\) starting with an initial position s(0) = 0. Find the displacement of the particle between t = 0 and t = 2, which is given by \(s(t)=\int_{0}^{2} v(t) d t\). Find the distance traveled by the particle during this interval, which is \(\int_{0}^{2}|v(t)| d t\).

Average height A baseball is launched into the outfield on a parabolic trajectory given by y = 0.01x(200 - x). Find the average height of the baseball during its flight.

Average values Find the average value of the functions shown in (a) and (b). Integration is not needed.

An unknown function The function f satisfies the equation \(3 x^{4}-2=\int_{a}^{x} f(t) d t\). Find f and check your answer by substitution.

An unknown function Assume \(f^{\prime}\) is a continuous function, \(\int_{1}^{2} f^{\prime}(2 x) d x=10\), and f(2) = 4. Evaluate f(4).

Function defined by an integral Let \(H(x)=\int_{0}^{x} \sqrt{4-t^{2}} d t\). a. Evaluate H(0). b. Evaluate \(H^{\prime}(1)\). c. Evaluate \(H^{\prime}(2)\). d. Use geometry to evaluate H(2). e. Find the value of s such that H(x) = sH(-x).

Function defined by an integral Make a graph of the function \(f(x)=\int_{1}^{x} \frac{d t}{t}\) for \(x \geq 1\). Be sure to include all of the evidence you used to arrive at the graph.

Identifying functions Match the graphs A, B, and C in the figure with the functions f(x), \(f^{\prime}(x)\), and \(\int_{0}^{x} f(t) d t\).

Geometry of integrals Without evaluating the integrals, explain why the following statement is true for positive integers n : \(\int_{0}^{1} x^{n} d x+\int_{0}^{1} \sqrt[n]{x} d x=1\).

Change of variables Use the change of variables \(u^{3}=x^{2}-1\) to evaluate the integral \(\int_{1}^{3} x \sqrt[3]{x^{2}-1} d x\).

Inverse tangent integral Prove that, for nonzero constants a and b, \(\int \frac{d x}{a^{2} x^{2}+b^{2}}=\frac{1}{a b} \tan ^{-1}\left(\frac{a x}{b}\right)\).

Additional integrals Evaluate the following integrals. \(\int \frac{\sin 2 x}{1+\cos ^{2} x} d x \quad(\sin 2 x=2 \sin x \cos x)\)

Additional integrals Evaluate the following integrals. \(\int \frac{1}{x^{2}} \sin \left(\frac{1}{x}\right) d x\)

Additional integrals Evaluate the following integrals. \(\int \frac{\left(\tan ^{-1} x\right)^{5}}{1+x^{2}} d x\)

Additional integrals Evaluate the following integrals. \(\int \frac{d x}{\left(\tan ^{-1} x\right)\left(1+x^{2}\right)}\)

Additional integrals Evaluate the following integrals. \(\int \frac{\sin ^{-1} x}{\sqrt{1-x^{2}}} d x\)

Additional integrals Evaluate the following integrals. \(\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x\)

Area with a parameter Let a > 0 and consider the family of functions f(x) = sin ax on the interval \([0, \pi / a]\). a. Graph f for a = 1, 2, 3. b. Let g(a) be the area of the region bounded by the graph of f and the x-axis on the interval \([0, \pi / a]\). Graph g for \(0<a<\infty\). Is g an increasing function, a decreasing function, or neither?

Equivalent equations Explain why a function that satisfies the equation \(u(x)+2 \int_{0}^{x} u(t) d t=10\) also satisfies the equation \(u^{\prime}(x)+2 u(x)=0\).

Area function properties Consider the function \(f(x)=x^{2}-5 x+4\) and the area function \(A(x)=\int_{0}^{x} f(t) d t\). a. Graph f on the interval [0, 6]. b. Compute and graph A on the interval [0, 6]. c. Show that the local extrema of A occur at the zeros of f. d. Give a geometrical and analytical explanation for the observation in part (c). e. Find the approximate zeros of A, other than 0 , and call them \(x_{1}\) and \(x_{2}\). f. Find b such that the area bounded by the graph of f and the x-axis on the interval \(\left[0, x_{1}\right]\) equals the area bounded by the graph of f and the x-axis on the interval \(\left[x_{1}, b\right]\). g. If f is an integrable function and \(A(x)=\int_{a}^{x} f(t) d t\), is it always true that the local extrema of A occur at the zeros of f? Explain.

Function defined by an integral Let \(f(x)=\int_{0}^{x}(t-1)^{15}(t-2)^{9} d t\). a. Find the intervals on which f is increasing and the intervals on which f is decreasing. b. Find the intervals on which f is concave up and the intervals on which f is concave down. c. For what values of x does f have local minima? Local maxima? d. Where are the inflection points of f?

Exponential inequalities Sketch a graph of \(f(t)=e^{t}\) on an arbitrary interval [a, b]. Use the graph and compare areas of regions to prove that \(e^{(a+b) / 2}<\frac{e^{b}-e^{a}}{b-a}<\frac{e^{a}+e^{b}}{2}\). (Source: Mathematics Magazine 81, no. 5 (December 2008): 374)