Textbook Question
{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.
b) Calculate g'(π)
g(π) = β«βΛ£ sinΒ² t dt
1
views
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 5.3.102b
{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.
(b) Calculate g'(π)
g(π) = β«βΛ£ sin (ΟtΒ² ) dt ( a Fresnel integral)
Verified step by step guidance1
Step 1: Recognize that the function g(π) is defined as a definite integral with a variable upper limit. This is a classic application of the Fundamental Theorem of Calculus, which states that if g(π) = β«βΛ£ f(t) dt, then g'(π) = f(π), provided f is continuous.
Step 2: Identify the integrand of g(π). In this case, the integrand is sin(ΟtΒ²). According to the Fundamental Theorem of Calculus, g'(π) will be equal to the integrand evaluated at the upper limit of integration, which is π.
Step 3: Substitute the upper limit π into the integrand. This means g'(π) = sin(ΟπΒ²).
Step 4: Confirm that the derivative g'(π) does not require further simplification, as the integrand sin(ΟπΒ²) is already expressed in terms of π.
Step 5: Note that no additional integration or differentiation is needed, as the problem specifically asks for g'(π), which is directly obtained using the Fundamental Theorem of Calculus.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus connects differentiation and integration, stating that if a function is defined as an integral with a variable upper limit, its derivative can be found by evaluating the integrand at that upper limit. Specifically, if g(x) = β«βΛ£ f(t) dt, then g'(x) = f(x). This theorem is essential for calculating the derivative of functions defined by integrals.
Recommended video:
06:11
Fundamental Theorem of Calculus Part 1
Definite Integral
A definite integral represents the accumulation of quantities, such as area under a curve, between two specified limits. In the context of the given function g(x) = β«βΛ£ sin(ΟtΒ²) dt, the integral computes the area under the curve of sin(ΟtΒ²) from 0 to x. Understanding how to evaluate definite integrals is crucial for applying the Fundamental Theorem of Calculus.
Recommended video:
05:43Definition of the Definite Integral
Fresnel Integral
The Fresnel integral is a specific type of integral that arises in wave optics and is defined as g(x) = β«βΛ£ sin(ΟtΒ²) dt. It is important in various applications, including diffraction and interference patterns. Recognizing the properties and behavior of Fresnel integrals helps in understanding the function g(x) and its derivative.
Recommended video:
06:18Integration by Parts for Definite Integrals
Related Practice
Textbook Question
Using properties of integrals Use the value of the first integral I to evaluate the two given integrals.
I = β«βΒΉ (πΒ³ β 2π) dπ = β3/4
(b) β«ββ° (2πβπΒ³) dπ
Textbook Question
Properties of integrals Use only the fact that β«ββ΄ 3π (4 βπ) dπ = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(b) β«ββ΄ π(π β 4) d(π)
Textbook Question
The following functions are positive and negative on the given interval.
Ζ(π) = xeβ»Λ£ on [-1,1]
(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.
1
views
Textbook Question
{Use of Tech} Riemann sums for larger values of n Complete the following steps for the given function f and interval.
Ζ(π) = xΒ² β 1 on [2,5] ; n = 75
(b) Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f and the x-axis on the interval.
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(b) A left Riemann sum always overestimates the area of a region bounded by a positive increasing function and the x-axis on an interval [a,b].
1
views