/*! 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 37 Evaluate the integral. $$ \i... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 37

Evaluate the integral. $$ \int_{0}^{1}(3-2 x)^{4} d x $$

Short Answer

Expert verified
The value of the given integral \(\int_{0}^{1}(3-2 x)^{4} d x\) is \(121\frac{2}{5}\), obtained by performing integration by substitution \(u = 3 - 2x\), finding the derivative of u with respect to x, substituting dx, changing limits of integration, and then evaluating the integral with the new limits.

Step by step solution

01

Perform substitution

Let \( u = 3 - 2x \). Now we will find the derivative of u with respect to x, or du/dx. $$ \frac{du}{dx} = -2 $$ Now, we will solve for dx and substitute it into our integral equation. $$ dx = \frac{du}{-2} $$ Let's change the limits of integration according to the substitution. When x = 0, $$ u = 3 - 2(0) = 3 $$ When x = 1, $$ u = 3 - 2(1) = 1 $$
02

Substitute and integrate

Now we will substitute u, and dx into the integral equation and also change the limits of integration accordingly. $$ \int_{3}^{1} u^{4} \frac{du}{-2} $$ Now, integrate with respect to u. $$ - \frac{1}{2} \int_{3}^{1} u^4 du = - \frac{1}{10} [u^5]_{3}^{1} $$
03

Evaluate the integral with new limits

Now we will evaluate the integral by substituting the new limits of integration (3 and 1). $$ - \frac{1}{10} [(1)^5 - (3)^5] = -\frac{1}{10} [1-243] $$
04

Simplify and obtain the final answer

Now we will simplify the expression to get the final value of the integral. $$ - \frac{1}{10}[-242] = \frac{242}{10} = 121\frac{2}{5} $$ So, the value of the given integral is \( 121\frac{2}{5} \).

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Key Concepts

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

Definite Integrals
In calculus, a definite integral is a way to calculate the area under a curve defined by a function over a specific interval.

Unlike indefinite integrals, which represent a family of functions, definite integrals have limits of integration鈥攖wo specific values that show where the calculation begins and ends.

This process involves:
  • Finding the antiderivative of the function.
  • Evaluating it at the upper limit of integration.
  • Subtracting the antiderivative evaluated at the lower limit.

The result gives the "net area" between the curve and the x-axis, taking into account regions above and below the axis.

For example, in our original problem \(\int_{0}^{1}(3-2 x)^{4} dx\), the integration took place from \(x = 0\) to \(x = 1\), resulting in a specific numerical value.
Change of Variables
Change of variables, also known as integration by substitution, is a powerful method for simplifying integrals.

It involves substituting a part of the integral with a new variable, typically denoted as \(u\). This technique can transform a complex integral into a simpler form that is easier to evaluate.

Here's how it works:
  • Identify a substitution \(u = f(x)\) that simplifies the integral.
  • Calculate \(du/dx\) to find the differential \(dx\).
  • Replace \(dx\) in the integral with \(du\).
  • Change the limits of integration if dealing with a definite integral.

In our exercise, we performed the substitution \(u = 3 - 2x\), transforming the integral into \(\int u^4 \, (-\frac{1}{2}) \, du\), simplifying calculations significantly.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration, establishing a profound link between the two. It has two primary parts:
  • First part: It states that the integral of a function over an interval can be reversed by taking its derivative.
  • Second part: If you have an antiderivative \(F(x)\) of a continuous function \(f(x)\), then the definite integral from \(a\) to \(b\) is found using \(F(b) - F(a)\).

This theorem explains why the process of finding the area under a curve (integration) relates closely to finding rates of change (differentiation).

In our example, after changing variables, we found the antiderivative of \(u^4\) and then applied the new limits to compute the definite integral, thus harnessing the power of the Fundamental Theorem.

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