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5.3. INTEGRATION BY SUBSTITUTION
paragraph to discuss the similarities among the three indefinite integrals in this
problem and the role of substitution and algebraic rearrangement in each.
3. Consider the indefinite integral
sin
3 (x) dx.
(a) Explain why the substitution u = sin(x) will not work to help evaluate the given
integral.
(b) Recall the Fundamental Trigonometric Identity, which states that sin
2 (x) +
cos 2 (x) = 1. By observing that sin
3 (x) = sin(x) · sin
2 (x), use the Fundamental
Trigonometric Identity to rewrite the integrand as the product of sin(x) with
another function.
(c) Explain why the substitution u = cos(x) now provides a possible way to evaluate
the integral in (b).
(d) Use your work in (a)-(c) to evaluate the indefinite integral
sin
3 (x) dx.
(e) Use a similar approach to evaluate
cos
3 (x) dx.
4. For the town of Mathland, MI, residential power consumption has shown certain trends
over recent years. Based on data reflecting average usage, engineers at the power
company have modeled the town’s rate of energy consumption by the function
r(t) = 4 + sin(0.263t + 4.7) + cos(0.526t + 9.4).
Here, t measures time in hours after midnight on a typical weekday, and r is the rate of
consumption in megawatts 5 at time t. Units are critical throughout this problem.
(a) Sketch a carefully labeled graph of r(t) on the interval [0,24] and explain its
meaning. Why is this a reasonable model of power consumption?
(b) Without calculating its value, explain the meaning of
24
0
r(t) dt. Include
appropriate units on your answer.
(c) Determine the exact amount of power Mathland consumes in a typical day.
(d) What is Mathland’s average rate of energy consumption in a given 24-hour
period? What are the units on this quantity?
5 The unit megawatt is itself a rate, which measures energy consumption per unit time. A megawatt-hour is
the total amount of energy that is equivalent to a constant stream of 1 megawatt of power being sustained for
1 hour.
5.3. INTEGRATION BY SUBSTITUTION
paragraph to discuss the similarities among the three indefinite integrals in this
problem and the role of substitution and algebraic rearrangement in each.
3. Consider the indefinite integral
sin
3 (x) dx.
(a) Explain why the substitution u = sin(x) will not work to help evaluate the given
integral.
(b) Recall the Fundamental Trigonometric Identity, which states that sin
2 (x) +
cos 2 (x) = 1. By observing that sin
3 (x) = sin(x) · sin
2 (x), use the Fundamental
Trigonometric Identity to rewrite the integrand as the product of sin(x) with
another function.
(c) Explain why the substitution u = cos(x) now provides a possible way to evaluate
the integral in (b).
(d) Use your work in (a)-(c) to evaluate the indefinite integral
sin
3 (x) dx.
(e) Use a similar approach to evaluate
cos
3 (x) dx.
4. For the town of Mathland, MI, residential power consumption has shown certain trends
over recent years. Based on data reflecting average usage, engineers at the power
company have modeled the town’s rate of energy consumption by the function
r(t) = 4 + sin(0.263t + 4.7) + cos(0.526t + 9.4).
Here, t measures time in hours after midnight on a typical weekday, and r is the rate of
consumption in megawatts 5 at time t. Units are critical throughout this problem.
(a) Sketch a carefully labeled graph of r(t) on the interval [0,24] and explain its
meaning. Why is this a reasonable model of power consumption?
(b) Without calculating its value, explain the meaning of
24
0
r(t) dt. Include
appropriate units on your answer.
(c) Determine the exact amount of power Mathland consumes in a typical day.
(d) What is Mathland’s average rate of energy consumption in a given 24-hour
period? What are the units on this quantity?
5 The unit megawatt is itself a rate, which measures energy consumption per unit time. A megawatt-hour is
the total amount of energy that is equivalent to a constant stream of 1 megawatt of power being sustained for
1 hour.