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Integrals and Sigma
Hello I need help with 2 problems.
The first is
Code:
F(x)= Integral from sqrt(x) to 2 (tan(t)/t^2) dt Find F'(x)
The second is
Code:
Find the actual area under the curve f(x)=2x^2 on the interval [0,3]. Use right hand endpoints and the formula
Area = lim n→ ∞ n Σ i=1 f(x^*i) delta(x)
Here is an image of the equations
Attachment 26043

Re: Integrals and Sigma
For the first part, I would perform the following transformation:
$\displaystyle \displaystyle \begin{align*} \int_2^{\sqrt{x}}{\frac{\tan{(t)}}{t^2}\,dt} &= \frac{1}{2}\int_2^{\sqrt{x}}{\frac{2t\tan{(t)}}{t^ 3}\,dt} \end{align*}$
Let $\displaystyle \displaystyle \begin{align*} u = t^2 \implies du = 2t\,dt \end{align*}$ and changing the bounds gives
$\displaystyle \displaystyle \begin{align*} \frac{1}{2} \int_4^x { \frac{ \tan{ \left( \sqrt{u} \right) } }{ u^{\frac{3}{2}} }\,du } \end{align*}$
Then you can make use of the Second Fundamental Theorem of Calculus.
For the second, you have been given a formula to use. Where exactly are you stuck?

Re: Integrals and Sigma
Okay thanks ill take another look at the first problem and for the second problem the equation confuses me. What does the x*i mean?

Re: Integrals and Sigma
$\displaystyle \displaystyle \begin{align*} x^*_i \end{align*}$ is just a shorthand to give a label to each value of x.

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Re: Integrals and Sigma
Okay I get problem 2 all the way up to this point
Attachment 26047
What is the n^3/3 + ...