Math Problems Vault

Problem 2531

Using the rectangle formula ( $n=12$ ), approximately compute

\int _{0}^{2\pi }x\sin x dx

and compare the result with the exact value.

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Problem 2532

Use the trapezoid formula to compute the following integral and estimate its error:

\int _{0}^{1}\dfrac {dx}{1+x}\quad (n=8) .

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Problem 2533

Use the trapezoid formula to compute the following integral and estimate its error:

\int _{0}^{1}\dfrac {dx}{1+x^{3}}\quad (n=12) .

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Problem 2534

Use the trapezoid formula to compute the following integral and estimate its error:

\int _{0}^{\frac{\pi }{2}}\sqrt{1-\dfrac {1}{4}\sin ^{2}x} dx\quad (n=6) .

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Problem 2535

Use Simpson's formula to compute the following integral:

\int _{1}^{9}\sqrt{x} dx\quad (n=4) .

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Problem 2536

Use Simpson's formula to compute the following integral:

\int _{0}^{\pi }\sqrt{3+\cos x} dx\quad (n=6) .

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Problem 2537

Use Simpson's rule to compute the following integral:

\int _{0}^{\frac{\pi }{2}}\dfrac {\sin x}{x} dx\quad (n=10) .

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Problem 2538

Use Simpson's rule to compute the following integral:

\int _{0}^{1}\dfrac {x dx}{\ln (1+x)}\quad (n=6) .

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Problem 2539

Taking $n=10$ , compute the Gauss constant (Catalan constant)

C=\int _{0}^{1}\dfrac {\arctan x}{x} dx .

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Problem 2540

Use the formula

\dfrac {\pi }{4}=\int _{0}^{1}\dfrac {dx}{1+x^{2}}

to compute the number $\pi$ accurate to $10^{-5}$ .

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Problem 2541

Compute

\int _{0}^{1}e^{x^{2}} dx

accurate to $0.001$ .

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Problem 2542

Compute

\int _{0}^{1}(e^{x}-1)\ln \dfrac {1}{x} dx

accurate to $10^{-4}$ .

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Problem 2543

Approximately compute the probability integral

\int _{0}^{+\infty }e^{-x^{2}} dx .

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Problem 2544

Approximately find the perimeter of an ellipse with semi-axes $a=10$ and $b=6$ .

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Problem 2545

Taking $\Delta x=\dfrac {\pi }{3}$ , plot pointwise the graph of the function

y=\int _{0}^{x}\dfrac {\sin t}{t} dt\quad (0\leqslant x\leqslant 2\pi )

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