Suppose that int_{0}^{1} f(t)dt =7. Calculate each of the following. a. int_{0}^{1} f(10t)dt = b....

Question:

Suppose that {eq}\int_{0}^{1} f(t)dt =7 {/eq}. Calculate each of the following.

{eq}a. \int_{0}^{1} f(10t)dt =\\ b. \int_{0}^{1} f(1-10t)dt =\\ c. \int_{0}^{1} f(5-4t)dt = {/eq}

Substitution with Unknown Function:

If we have an integer value of the given definite integral expression with unknown function, then we'll rearrange all the other definite integral expressions as the given expression using the substitution method.

For that, we'll substitute the variable of the unknown function by the variable {eq}u {/eq} and differentiate the substitution to replace the {eq}dt {/eq} term of the given integral expression.

Answer and Explanation:

a.

The given value of the definite

{eq}\int_{0}^{1} f(t)dt =7 {/eq}

We can write the above expression as:

{eq}\int_{0}^{1} f(u) \ du =7 {/eq}

{eq}\int_{0}^{1} f(10t)dt =? {/eq}


Take {eq}u=10t {/eq} in the above expression by the substitution method of the integral.

The differentiation of both sides of the above expression with their respective variables is:

{eq}\displaystyle \ du = 10 \ dt {/eq}

Using the above values, we'll simplify the indefinite integral as:

{eq}\begin{align*} \displaystyle \int f(10t)dt &= \int \frac{1}{10}f(u)\ du\\ &= \frac{1}{10} \int f(u)\ du\\ \end{align*} {/eq}


Substitute all the above values in the definite integral expression and simplify it.

{eq}\begin{align*} \displaystyle \int_{0}^{1} f(10t)dt &= \frac{1}{10} \int_{0}^{1} f(u)\ du\\ &= \frac{1}{10} (7)\\ &=\frac{7}{10} \end{align*} {/eq}


b.

The given definite integral expression is:

{eq}\int_{0}^{1} f(1-10t)dt =? {/eq}

For the substitution, we'll take {eq}1-10t=u {/eq}

Differentiation of both sides of the above substitution is:

{eq}\begin{align*} \displaystyle 0-10(1)\ dt& =\ du\\ -10\ dt& =\ du\\ \ dt& =-\frac{1}{10}\ du\\ \end{align*} {/eq}


From the above values, we have:

{eq}\displaystyle \int f(u)\left ( -\frac{1}{10}\ du \right )=-\frac{1}{10}\int f(u) \ du \\ {/eq}

Now, the value of the definite integral using the above and given values is:

{eq}\begin{align*} \displaystyle -\frac{1}{10}\int_{0}^{1} f(u) \ du &=-\frac{1}{10}(7)\\ &=-\frac{7}{10}\\ \end{align*} {/eq}


c.

The given definite integral expression is:

{eq}\int_{0}^{1} f(5-4t)dt =? {/eq}

Using the substitution method, we'll take {eq}5-4t {/eq} as the varaiable {eq}u {/eq}.

The differentiation of the above substitution expression is:

{eq}\begin{align*} \displaystyle 0-4(1)\ dt &=\ du\\ \ dt &=-\frac{1}{4}\ du\\ \end{align*} {/eq}


Substituting the above values in the indefinite integral expression, we get:

{eq}\begin{align*} \displaystyle \int f(5-4t)dt &=\int f(u)\left ( -\frac{1}{4}\ du \right )\\ &=-\frac{1}{4}\int f(u)\ du \\ \end{align*} {/eq}


Now, the value of the given definite integral expression is:

{eq}\begin{align*} \displaystyle \int_{0}^{1} f(5-4t)dt&=-\frac{1}{4}\int_{0}^{1} f(u)\ du \\ &=-\frac{1}{4}(7) \\ &=-\frac{7}{4} \end{align*} {/eq}


Learn more about this topic:

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How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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