Textbook Question
Use geometry and properties of integrals to evaluate
β«βΒΉ (2π + β(1βπΒ²) + 1) dπ
Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.2.61
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of Ζ and the π-axis. Evaluate the following integrals.
β«βαΆ Ζ(π) dπ
Verified step by step guidance1
Identify the points a and c on the x-axis from the graph, which define the limits of integration for the integral \( \int_{a}^{c} f(x) \, dx \).
Observe the graph of \( f(x) \) between \( x = a \) and \( x = c \) and note the areas bounded by the curve and the x-axis. Pay attention to whether the graph is above or below the x-axis in each region, as this affects the sign of the integral.
Calculate the area of each region between the curve and the x-axis separately. If the graph is above the x-axis, the area contributes positively to the integral; if below, it contributes negatively.
Sum the signed areas of all regions between \( a \) and \( c \) to find the value of the definite integral \( \int_{a}^{c} f(x) \, dx \).
Express the final integral as the algebraic sum of these areas, which represents the net area between the curve and the x-axis over the interval \( [a, c] \).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral as Net Area
A definite integral represents the net area between the graph of a function and the x-axis over a given interval. Areas above the x-axis contribute positively, while areas below contribute negatively, affecting the integral's value.
Recommended video:
05:43
Definition of the Definite Integral
Interpreting Graphical Areas
When evaluating integrals from graphs, it is essential to identify and measure the areas of regions bounded by the curve and the x-axis. Understanding which parts lie above or below the axis helps determine the sign and magnitude of each area.
Recommended video:
05:06Finding Area When Bounds Are Not Given
Properties of Definite Integrals
Definite integrals are additive over adjacent intervals, meaning the integral from a to c can be split into integrals over subintervals. This property allows summing individual areas to find the total integral value.
Recommended video:
05:43Definition of the Definite Integral
Related Practice
Textbook Question
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found.
β«ββ΅ (πΒ²β9) dπ
Textbook Question
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« πeΛ£Β² dπ
Textbook Question
Areas of regions Find the area of the following regions.
The region bounded by the graph of Ζ(π) = x /β(πΒ² β9) and the π-axis between and π = 4 and π= 5
1
views
Textbook Question
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« 2π(πΒ² β 1)βΉβΉ dπ
Textbook Question
Average velocity The velocity in m/s of an object moving along a line over the time interval [0,6] is v (t) = tΒ² + 3t. Find the average velocity of the object over this time interval.