Chapter 5

Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch. \( \displaystyle \int^{2\pi}_{\pi/6} \cos x \,dx \)

Suppose \( f \) has absolute minimum value \( m \) and absolute maximum value \( M \). Between what two values must \( \displaystyle \int^2_0 f(x) \,dx \) lie? Which property of integrals allows you to make your conclusion?

Evaluate the definite integral. \( \displaystyle \int^1_0 \sqrt[3]{1 + 7x} \, dx \)

In Section 4.7 we defined the marginal revenue function \( R'(x) \) as the derivative of the revenue function \( R(x) \), where \( x \) is the number of units sold. What does \( \displaystyle \int^{5000}_{1000} R'(x) \, dx \) represent?

What is wrong with the equation? \( \displaystyle \int^1_{-2} x^{-4} \, dx = \frac{x^{-3}}{-3} \Bigg]^1_{-2} = -\frac{3}{8} \)

Use the properties of integrals to verify the inequality without evaluating the integrals. \( \displaystyle \int^4_0 (x^2 - 4x + 4) \,dx \ge 0 \)

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts \( (a) - (d) \) to sketch the graph of \( f \). \( f(x) = \ln (1 - \ln x) \)

Evaluate the definite integral. \( \displaystyle \int^3_0 \frac{dx}{5x + 1} \)

If \( f(x) \) is the slope of a trail at a distance of \( x \) miles from the start of the trail, what does \( \displaystyle \int^5_3 f(x) \, dx \) represent?

What is wrong with the equation? \( \displaystyle \int^2_{-1} \frac{4}{x^3} \, dx = -\frac{2}{x^2} \Bigg]^2_{-1} = -\frac{3}{2} \)

Evaluate the definite integral. \( \displaystyle \int^{\pi/6}_0 \frac{\sin t}{\cos^ 2 t} \, dt \)

If \( x \) is measured in meters and \( f(x) \) is measured in newtons, what are the units for \( \displaystyle \int^{100}_0 f(x) \, dx \)?

What is wrong with the equation? \( \displaystyle \int^{\pi}_{\pi/3} \sec \theta \tan \theta \, d\theta = \sec \theta \Bigg]^{\pi}_{\pi/3} = -3 \)

Use the properties of integrals to verify the inequality without evaluating the integrals. \( \displaystyle 2 \le \int^1_{-1} \sqrt{1 + x^2} \,dx \le 2\sqrt{2} \)

Evaluate the definite integral. \( \displaystyle \int^{2\pi/3}_{\pi/3} \csc^2 \biggl( \frac{1}{2}t \biggr) \, dt \)

If the units for \( x \) are feet and the units for \( a(x) \) are pounds per foot, what are the units for \( da/dx \)? What units does \( \displaystyle \int^8_2 a(x) \, dx \) have?

What is wrong with the equation? \( \displaystyle \int^{\pi}_{0} \sec^2 x \, dx = \tan x \Bigg]^{\pi}_{0} = 0 \)

Use the properties of integrals to verify the inequality without evaluating the integrals. \( \displaystyle \frac{\pi}{12} \le \int^{\pi/3}_{\pi/6} \sin x \,dx \le \frac{\sqrt{3}\pi}{12} \)

Evaluate the definite integral. \( \displaystyle \int^2_1 \frac{e^{1/x}}{x^2} \, dx \)

The velocity function (in meters per second) is given for a particle moving along a line. Find (a) the displacement and (b) the distance traveled by the particle during the given time interval. \( v(t) = 3t - 5 \), \( 0 \le t \le 3 \)

Find the derivative of the function. \( g(x) = \displaystyle \int^{3x}_{2x} \frac{u^2 - 1}{u^2 + 1} \, du \) \( \displaystyle \Biggl[ Hint: \int^{3x}_{2x} f(u) \, du = \int^0_{2x} f(u) \, du + \int^{3x}_0 f(u) \, du \Biggr] \)

Evaluate the definite integral. \( \displaystyle \int^1_0 xe^{-x^2} \, dx \)

The velocity function (in meters per second) is given for a particle moving along a line. Find (a) the displacement and (b) the distance traveled by the particle during the given time interval. \( v(t) = t^2 - 2t - 3 \), \( 2 \le t \le 4 \)

Find the derivative of the function. \( g(x) = \displaystyle \int^{1 + 2x}_{1 - 2x} t \sin t \, dt \)

Evaluate the definite integral. \( \displaystyle \int^{\pi/4}_{-\pi/4} (x^3 + x^4 \tan x) \, dx \)

The acceleration function (in \( m/s^2 \)) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time \( t \) and (b) the distance traveled during the given time interval. \( a(t) = t + 4 \), \( v(0) = 5 \), \( 0 \le t \le 10 \)

Find the derivative of the function. \( F(x) = \displaystyle \int^{x^2}_{x} e^{t^2} \, dt \)

Evaluate the definite integral. \( \displaystyle \int^{\pi/2}_{0} \cos x \sin(\sin x) \, dx \)

The acceleration function (in \( m/s^2 \)) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time \( t \) and (b) the distance traveled during the given time interval. \( a(t) = 2t + 3 \), \( v(0) = -4 \), \( 0 \le t \le 3 \)

Find the derivative of the function. \( F(x) = \displaystyle \int^{2x}_{\sqrt{x}} \arctan t \, dt \)

Evaluate the definite integral. \( \displaystyle \int^{13}_0 \frac{dx}{\sqrt[3]{(1 + 2x)^2}} \)

The linear density of a rod of length \( 4 m \) is given by \( \rho (x) = 9 + 2 \sqrt{x} \) measured in kilograms per meter, where \( x \) is measured in meters from one end of the rod. Find the total mass of the rod.

Find the derivative of the function. \( y = \displaystyle \int^{\sin x}_{\cos x} \ln (1 + 2v) \, dv \)

Evaluate the definite integral. \( \displaystyle \int^a_0 x\sqrt{a^2 - x^2} \, dx \)

Water flows from the bottom of a storage tank at a rate of \( r(t) = 200 - 4t \) liters per minute, where \( 0 \le t \le 50 \). Find the amount of water that flows from the tank during the first 10 minutes.

If \( \displaystyle f(x) = \int^x_0 (1 - t^2) e^{t^2} \,dt \), on what interval is \( f \) increasing?

Evaluate the definite integral. \( \displaystyle \int^a_0 x\sqrt{x^2 + a^2} \, dx \) \( (a > 0) \)

Evaluate the definite integral. \( \displaystyle \int^{\pi/3}_{-\pi/3} x^4 \sin x \, dx \)

Evaluate the definite integral. \( \displaystyle \int^2_1 x\sqrt{x - 1} \, dx \)

The marginal cost of manufacturing \( x \) yards of a certain fabric is $$ C'(x) = 3 - 0.01x + 0.000006x^2 $$ (in dollars per yard). Find the increase in cost if the production level is raised from 2000 yards to 4000 yards.

Let \( \displaystyle F(x) = \int^x_2 e^{t^2} \, dt \). Find an equation of the tangent line to the curve \( y = F(x) \) at the point with \( x \)-coordinate 2.

Which of the integrals \( \displaystyle \int^2_1 \arctan x \,dx \), \( \displaystyle \int^2_1 \arctan \sqrt{x} \,dx \), and \( \displaystyle \int^2_1 \arctan (\sin x) \,dx \) has the largest value? Why?

Evaluate the definite integral. \( \displaystyle \int^4_0 \frac{x}{\sqrt{1 + 2x}} \, dx \)

If \( \displaystyle f(x) = \int^{\sin x}_0 \sqrt{1 + t^2} \, dt \) and \( \displaystyle g(y) = \int^y_3 f(x) \, dx \), find \( g''(\pi/6) \).

Which of the integrals \( \displaystyle \int^{0.5}_0 \cos (x^2) \,dx \), \( \displaystyle \int^{0.5}_0 \cos \sqrt{x} \,dx \) is larger? Why?

Evaluate the definite integral. \( \displaystyle \int^{e^4}_{e} \frac{dx}{x \sqrt{\ln x}} \)

If \( f(1) = 12 \), \( f' \) is continuous, and \( \displaystyle \int^4_1 f'(x) \, dx = 17 \), what is the value of \( f(4) \)?

Evaluate the definite integral. \( \displaystyle \int^2_0 (x - 1)e^{(x - 1)^2} \, dx \)

The error function $$ \displaystyle \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int^x_0 e^{-t^2} \, dt $$ is used in probability, statistics, and engineering. (a) Show that \( \displaystyle \int^b_a e^{-t^2} \, dt = \frac{1}{2} \sqrt{\pi} [ \text{erf}(b) - \text{erf}(a) ] \). (b) Show that the function \( y = e^{x^2} \text{erf}(x) \) satisfies the differential equation \( y' = 2xy + 2/\sqrt{\pi} \).

(a) If \( f \) is continuous on \( [a, b] \), show that $$ \biggl| \int^b_a f(x) \,dx \biggr| \le \int^b_a \bigl| f(x) \bigr| \,dx $$ [\( Hint: \) \( -\bigl| f(x) \bigr| \le f(x) \le \bigl| f(x) \bigr|. \)] (b) Use the result of part (a) to show that $$ \biggl| \int^{2\pi}_0 f(x) \sin 2x \,dx \biggr| \le \int^{2\pi}_0 \bigl| f(x) \bigr| \,dx $$