# Hey's questions at Yahoo Answers regarding solving for a limit of integration

• MHB
• MarkFL
In summary, the conversation is about two math questions involving integrals. The first question asks for the value of "b" when given a specific integral and the second question asks for the value of "a" when given another integral. The process for solving each question involves applying the anti-derivative form of the FTOC and using logarithmic and exponential functions. The expert also invites others to post their calculus questions in the forum.
MarkFL
Gold Member
MHB
Here are the questions:

1) If b > 1 and ∫2x^4 dx = 1 (from b= b and a =1) what would be the value of "b"? how do I solve for b?

2) If a < 4 and ∫2.3e^(1.4x) dx = 46 (from b = 4 and a = a) what would be the value of "a"? how do I solve for a?

Here is a link to the questions:

I have posted a link there to this topic so the OP can find my response.

Hello hey,

1.) We are given:

$$\displaystyle 2\int_1^b x^4\,dx=1$$ where $$\displaystyle 1<b$$

Applying the anti-derivative form of the FTOC on the left side, we have:

$$\displaystyle \frac{2}{5}\left[x^4 \right]_1^b=1$$

Multiply through by $$\displaystyle \frac{5}{2}$$ and complete the FTOC:

$$\displaystyle b^5-1=\frac{5}{2}$$

Add through by $1$, and take the fifth root of both sides:

$$\displaystyle b=\left(\frac{7}{2} \right)^{\frac{1}{5}}>1$$

2.) We are given

$$\displaystyle 2.3\int_a^4 e^{1.4x}\,dx=46$$ where $$\displaystyle a<4$$

Applying the anti-derivative form of the FTOC on the left side, we have:

$$\displaystyle \frac{23}{14}\left[e^{1.4x} \right]_a^4=46$$

Multiply through by $$\displaystyle \frac{14}{23}$$ and complete the FTOC:

$$\displaystyle e^{5.6}-e^{1.4a}=28$$

Arrange with the term containing $a$ on the left, and everything else on the right:

$$\displaystyle e^{1.4a}=e^{5.6}-28$$

Convert from exponential to logarithmic form and then divide through by $1.4$:

$$\displaystyle a=\frac{5}{7}\ln\left(e^{\frac{28}{5}}-28 \right)<4$$

I have used fractions rather than decimals equivalents. I just prefer this notation.

To hey and any other guests viewing this topic, I invite and encourage you to post other calculus questions here in our http://www.mathhelpboards.com/f10/ forum.

Best Regards,

Mark.

## 1) What is a limit of integration?

A limit of integration is a mathematical concept used in integral calculus. It represents the boundaries of the region over which the integral is being evaluated.

## 2) How do I solve for a limit of integration?

To solve for a limit of integration, you need to first determine the function being integrated and the boundaries of the region. Then, you can use various techniques such as substitution, integration by parts, or trigonometric substitution to evaluate the integral.

## 3) What are some common techniques for solving a limit of integration?

Some common techniques for solving a limit of integration include substitution, integration by parts, trigonometric substitution, and partial fractions. It is important to choose the appropriate technique based on the form of the integral.

## 4) Can you provide an example of solving a limit of integration?

Sure, let's say we have the integral of 2x dx from x=0 to x=4. We can use the power rule of integration to solve this by raising the power of x by 1 and dividing by the new power. In this case, we get x^2. Then, we plug in the boundaries and subtract to get 16-0=16. So, the solution is 16.

## 5) What are some tips for solving a difficult limit of integration?

Some tips for solving a difficult limit of integration include breaking up the integral into smaller parts, using trigonometric identities, and practicing with different techniques. It is also helpful to review the properties of integrals and familiarize yourself with common integrals and their solutions.

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