/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Evaluate the following iterated ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 6

Evaluate the following iterated integrals. $$\int_{1}^{2} \int_{0}^{1}\left(3 x^{2}+4 y^{3}\right) d y d x$$

Short Answer

Expert verified
Question: Evaluate the iterated integral $$\int_{1}^{2} \int_{0}^{1} (3x^2 + 4y^3) dy dx$$. Answer: 8

Step by step solution

01

Identify the inner integral bounds and the expression

First, we need to identify the expression with respect to y and its bounds. The expression is: $$3x^2 + 4y^3$$ and the bounds are from 0 to 1, i.e., y varies from 0 to 1.
02

Integrate with respect to y

Now, we will integrate the expression \((3x^2 + 4y^3)\) with respect to y from 0 to 1: $$\int_{0}^{1} (3x^2 + 4y^3) dy$$ This integral becomes: $$\int_{0}^{1} (4y^3) dy + \int_{0}^{1} (3x^2) dy$$
03

Evaluate the inner integral

Evaluating the above integrals, we get the following expressions: $$\left[\frac{4}{4}y^4\right]_0^1 + \left[3x^2 y\right]_0^1$$ After evaluating the limits, we get: $$1^4 - 0^4 + 3x^2(1) - 3x^2(0) = 1 + 3x^2$$
04

Integrate with respect to x

Now, we will integrate the expression \((\)1 + 3x^2$$)$ with respect to x from 1 to 2: $$\int_{1}^{2} (1 + 3x^2) dx$$
05

Evaluate the outer integral

Evaluating the above integral, we get the following expression: $$\left[x + \frac{3}{3} x^3\right]_1^2$$ After evaluating the limits, we get: $$(2 + 2^3) - (1 + 1^3) = 2 + 8 -1 -1 = 8$$ Thus, the value of the given iterated integral is 8.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Limiting center of mass \(A\) thin rod of length \(L\) has a linear density given by \(\rho(x)=\frac{10}{1+x^{2}}\) on the interval \(0 \leq x \leq L\). Find the mass and center of mass of the rod. How does the center of mass change as \(L \rightarrow \infty ?\)

Let \(D\) be the solid bounded by \(y=x, z=1-y^{2}, x=0\) and \(z=0 .\) Write triple integrals over \(D\) in all six possible orders of integration.

A disk of radius \(r\) is removed from a larger disk of radius \(R\) to form an earring (see figure). Assume the earring is a thin plate of uniform density. a. Find the center of mass of the earring in terms of \(r\) and \(R\) (Hint: Place the origin of a coordinate system either at the center of the large disk or at \(Q\); either way, the earring is symmetric about the \(x\) -axis.) b. Show that the ratio \(R / r\) such that the center of mass lies at the point \(P\) (on the edge of the inner disk) is the golden mean \((1+\sqrt{5}) / 2 \approx 1.618\)

General volume formulas Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume that \(a, b, c, r, R,\) and \(h\) are positive constants. Frustum of a cone Find the volume of a truncated solid cone of height \(h\) whose ends have radii \(r\) and \(R\).

\(A\) thin plate is bounded by the graphs of \(y=e^{-x}, y=-e^{-x}, x=0,\) and \(x=L .\) Find its center of mass. How does the center of mass change as \(L \rightarrow \infty ?\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.