/*! 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 23 Find the area under each of the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 23

Find the area under each of the given curves. \(y=(x-3)^{4} ; x=1\) to \(x=4\)

Short Answer

Expert verified
The area under the curve is 6.6.

Step by step solution

01

Identify the Integral

To find the area under the curve, compute the definite integral of the function from the given limits. The function is \( y = (x-3)^4 \) from \( x = 1 \) to \( x = 4 \).
02

Setup the Integral

Set up the definite integral with the limits 1 and 4: \[ \int_{1}^{4} (x - 3)^4 \, dx. \]
03

Find the Antiderivative

To find the antiderivative, use the power rule for integration. The integral of \( (x - 3)^4 \) is \(\frac{(x-3)^5}{5} \). Apply this to get: \[ \frac{(x-3)^5}{5} \].
04

Evaluate the Antiderivative at the Upper and Lower Limits

Next, evaluate the antiderivative at the upper limit (4) and the lower limit (1) and subtract the results: \[ \frac{(4-3)^5}{5} - \frac{(1-3)^5}{5} = \frac{1^5}{5} - \frac{(-2)^5}{5} = \frac{1}{5} - \frac{-32}{5}. \]
05

Simplify the Result

Simplify the expression: \[ \frac{1}{5} - \frac{-32}{5} = \frac{1}{5} + \frac{32}{5} = \frac{33}{5} = 6.6. \]

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

definite integral
A definite integral calculates the accumulation of quantities, like area under a curve, between set boundaries.
We denote this using the integral symbol with upper and lower limits.
For example, in our exercise, we use the limits from 1 to 4: \( \int_{1}^{4} (x - 3)^4 \, dx\).
The limits show where we start and stop calculating the area.
This process helps in determining exact numbers, making it a powerful tool in calculus.
integration
Integration is a fundamental concept in calculus.
Unlike differentiation, which deals with rates of change, integration focuses on accumulation.
We use it to find quantities like area, volume, and total distance.
In our exercise, we integrated the function \( (x - 3)^4 \) to find the area under the curve.
We setup the integral like this: \( \int_{1}^{4} (x - 3)^4 \, dx\).
This process transforms the function into its antiderivative, allowing us to evaluate it at specified limits.
power rule for integration
The power rule for integration is a simple yet crucial technique.
It helps to transform functions into their antiderivatives.
The rule for a function of the type \( x^n \) is given by: \( \int x^n dx = \frac{x^{n+1}}{n+1} \).
In our exercise, we applied the power rule to \( (x-3)^4 \).
Treating \( (x-3) \) as a single unit, follow the rule to get: \( \frac{(x-3)^5}{5} \).
This antiderivative is then evaluated at the given limits, ensuring accurate calculation of the area.

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

Determine \(\Delta x\) and the midpoints of the subintervals formed by partitioning the given interval into \(n\) subintervals. $$3 \leq x \leq 5 ; n=5$$

Evaluate the given integral. $$\int_{0}^{\ln 2} \frac{e^{x}+e^{-x}}{2} d x$$

Use a Riemann sum to approximate the area under the graph of \(f(x)\) on the given interval, with selected points as specified. \(f(x)=x^{3} ; 0 \leq x \leq 1, n=5\), right endpoints

Find the real number \(b>0\) so that the area under the graph of \(y=x^{3}\) from 0 to \(b\) is equal to 4 .

Find the area of the region between the curve and the \(x\) -axis. \(f(x)=1-x^{2}\), from \(-2\) to 2.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

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.