Textbook Question
If ∫²₋₂ 3ƒ(x) dx = 12, ∫⁵₋₂ ƒ(x) dx = 6, and ∫⁵₋₂ g(x) dx = 2, find the value of each of the following.
d. ∫⁵₋₂ (-πg(x)) dx
1
views
Ch. 5 - Integrals
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 5, Problem 5.PE.10b
If ∫₀² ƒ(x) dx = π, ∫₀² 7g(x) dx = 7, and ∫₀¹ g(x) dx = 2, find the value of each of the following.
b. ∫₁² g(x) dx
Verified step by step guidance1
Recall the property of definite integrals that allows us to split an integral over an interval into the sum of integrals over subintervals: \(\int_a^c f(x) \, dx = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx\) for any \(a < b < c\).
Apply this property to the integral \(\int_0^2 g(x) \, dx\), splitting it at \(x=1\): \(\int_0^2 g(x) \, dx = \int_0^1 g(x) \, dx + \int_1^2 g(x) \, dx\).
We are given \(\int_0^1 g(x) \, dx = 2\), so substitute this value into the equation: \(\int_0^2 g(x) \, dx = 2 + \int_1^2 g(x) \, dx\).
Next, use the information about \(\int_0^2 7g(x) \, dx = 7\). Since the integral of a constant times a function is the constant times the integral of the function, write \(7 \int_0^2 g(x) \, dx = 7\).
Divide both sides by 7 to find \(\int_0^2 g(x) \, dx = 1\). Substitute this back into the equation from step 3 to solve for \(\int_1^2 g(x) \, dx\): \(1 = 2 + \int_1^2 g(x) \, dx\), then isolate \(\int_1^2 g(x) \, dx\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral and Area Interpretation
A definite integral ∫_a^b f(x) dx represents the net area under the curve f(x) from x = a to x = b. It accumulates the values of the function over the interval, which can be used to find total quantities or changes between two points.
Recommended video:
05:43
Definition of the Definite Integral
Linearity of the Integral
The integral operator is linear, meaning ∫_a^b [c·f(x)] dx = c·∫_a^b f(x) dx for any constant c. This property allows us to factor constants out of integrals and simplify expressions involving scaled functions.
Recommended video:
07:17Linearization
Additivity of Integrals over Adjacent Intervals
The integral over an interval can be split into sums over subintervals: ∫_a^c f(x) dx = ∫_a^b f(x) dx + ∫_b^c f(x) dx for a < b < c. This helps find unknown integrals by subtracting known parts from the whole.
Recommended video:
05:56Additional Rules for Indefinite Integrals
Related Practice
Textbook Question
Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
√x + √y = 1, x = 0, y = 0
1
views
Textbook Question
Definite Integrals
In Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval, and the numbers cₖ are chosen from the subintervals of P.
n
lim ∑ (cos(cₖ/2)) ∆xₖ, where P is a partition of [-π, 0]
∥P∥→0 k = 1
1
views
Textbook Question
In Exercises 11–14, find the total area of the region between the graph of f and the x-axis.
ƒ(x) = 5 - 5x²/³, -1 ≤ x ≤ 8
Textbook Question
10 10
Suppose that Σ aₖ = -2 and Σ bₖ = 25. Find the value of
k = 1 k = 1
10
a. Σ aₖ/4
k = 1
1
views
Textbook Question
Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
y² = 4x, y = 4x - 2