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hey's questions at Yahoo! Answers regarding solving for a limit of integration

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MarkFL

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Feb 24, 2012
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Here are the questions:

Can someone please help me with these two math questions? *integrals*?

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:

Can someone please help me with these two math questions? *integrals*? - Yahoo! Answers

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

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Feb 24, 2012
13,775
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 Calculus forum.

Best Regards,

Mark.