Approximate integration formulas (rectangle, trapezoid, Simpson)

Evaluate the integral

to an accuracy of two decimal places; that is, find a number $I^{*}$ such that $\left|I-I^{*}\right| < 0.005$.

Suppose the area $S=S(x)$ of the cross-section of a solid perpendicular to the $Ox$ axis varies according to the following quadratic law:

where $A$, $B$, and $C$ are constants.

Prove that the volume of this solid equals

where $H=b-a$ (Simpson's formula).

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

and compare the result with the exact value.

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

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

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

Use Simpson's formula to compute the following integral:

Use Simpson's formula to compute the following integral:

Use Simpson's rule to compute the following integral:

Use Simpson's rule to compute the following integral:

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

Compute approximately the integral $J$: a) by the rectangle rule $(n = 1)$; b) by the trapezoidal rule $(n = 1)$; c) by Simpson's rule $(n = 2)$, and find the difference between the exact and approximate values, if:

Find $\delta = \left|J-J^{*}\right|$, where $J$ is the exact value of the integral and $J *$ is its approximate value, computed with step $h = 2$ by the formula: a) rectangle rule; b) trapezoidal rule; c) Simpson's rule. In each case find the difference between the right-hand side $\Delta_{0}$ in the estimates (5), (9), (12) $(k = 4)$ respectively, and $\delta (2),(3)$: $J = \int_{1}^{5} x^{4} d x$.

Find $\delta = \left|J-J^{*}\right|$, where $J$ is the exact value of the integral and $J *$ is its approximate value, computed with step $h = 2$ by the formula: a) rectangle rule; b) trapezoidal rule; c) Simpson's rule. In each case find the difference between the right-hand side $\Delta_{0}$ in the estimates (5), (9), (12) $(k = 4)$ respectively, and $\delta (2),(3)$: $J = \int_{-2}^{2} e^{x} d x$.

Compute approximately, with $n = 12$ (computing to four decimal places), the integral $\int_{0}^{2 \pi } x \sin x d x$:

Find approximately, with $n = 6$ (computing to four decimal places), the integral $\int_{0}^{1} \frac{\sin x}{x} d x$:

Compute approximately, with $n = 10$, the integral $\int_{0}^{1} \sqrt{1+x} d x$:

Compute approximately, using the rectangle rule with step $h$, the integrals:

Compute approximately, using the trapezoidal formula with step $h$, the integrals:

Compute approximately, using Simpson's formula with step $h$, the integrals:

Compute approximately, using Simpson's formula with $n = 10$:

Compute approximately, using Simpson's formula with $n = 4$ and $n = 8$, the integral:

Compute approximately, using Simpson's formula with $n = 4$ and $n = 8$, the integral:

For the given integral, estimate the error of the rectangle rule with step $h$:

For the given integral, estimate the error of the trapezoidal rule with step $h$:

For the given integral, estimate the error of Simpson's rule with step $h$:

Find the step $h$ for which the error of the approximate value of the given integral by the rectangle rule is not more than $\varepsilon$:

Find the step $h$ for which the error of the approximate value of the integral by the trapezoidal rule is not more than $\varepsilon$:

Find the step $h$ for which the error of the approximate value of the integral by Simpson's rule is not more than $\varepsilon$:

Compute, using the rectangle rule, the integral

for $n = 4$ and find the error of the result.

Compute approximately, using the rectangle rule with error not exceeding $10^{-2}$:

Compute approximately, using the trapezoidal rule with error not exceeding $10^{-2}$:

Compute approximately, using Simpson's rule with error not exceeding $\varepsilon$:

Compute approximately, with error not exceeding $\varepsilon$, the integral:

Find the surface area of a hemisphere of radius $R$, using Simpson's formula with $n = 2$.

Prove that Simpson's formula with $n = 2$ gives an exact result when computing the volume of:

Using Simpson's formula, compute, with error not exceeding $5 \cdot 10^{-2}$, the volume of the solid formed by rotating the curve $y = \sin x$, $0 \leqslant x \leqslant \pi$, around the axis $O x$.

Find approximately, using Simpson's formula, the length of the arc of the given curve with error not exceeding $10^{-8}$:

Compute the length of an ellipse with eccentricity $\varepsilon = 0,5$ and semi-axis $a = 1$, using Simpson's formula with $n = 6$. Estimate the error of the result.

Find approximately, with error not exceeding $5 \cdot 10^{-2}$, the length of an ellipse with semi-axes $a = 10$ and $b = 6$.

Find approximately, with error not exceeding 0.03, the surface area of the ellipsoid formed by rotating the ellipse $x^{2} + 4 y^{2} = 4$ around the axis $O x$ (use Simpson's formula).

Find approximately, with error not exceeding 0.01, the volume of the solid formed by rotating around the axis $O x$ the figure $0 \leqslant y \leqslant 1 /(1+x^{2}), \quad -2 \leqslant x \leqslant 2$.

Prove inequality (5) for the rectangle rule.

Here $[a ; b]$ is divided into $n$ equal parts, $h = (b-a) / n$, $x_{i} = a+(i+1 / 2) h$, $0 \leqslant i \leqslant n-1$, the rectangle rule is $\int_{a}^{b} f(x) d x \approx h \sum_{i = 0}^{n-1} f\left(x_{i}\right)$ and $\Delta = \int_{a}^{b} f(x) d x-h \sum_{i = 0}^{n-1} f\left(x_{i}\right)$ is its error. Inequality (5): if $f$ has a piecewise continuous second derivative on $[a ; b]$, then $\left|\Delta \right| \leqslant \frac{b-a}{24} h^{2} \sup_{[a ; b]}\left|f^{\prime \prime }(x)\right|$.

Prove that inequality (5) is sharp, i.e. there exists a function having a piecewise continuous second derivative such that

Here $[a ; b]$ is divided into $n$ equal parts, $h = (b-a) / n$, $x_{i} = a+(i+1 / 2) h$, $0 \leqslant i \leqslant n-1$, and $\Delta = \int_{a}^{b} f(x) d x-h \sum_{i = 0}^{n-1} f\left(x_{i}\right)$ is the error of the rectangle rule $\int_{a}^{b} f(x) d x \approx h \sum_{i = 0}^{n-1} f\left(x_{i}\right)$. Inequality (5): if $f$ has a piecewise continuous second derivative on $[a ; b]$, then $\left|\Delta \right| \leqslant \frac{b-a}{24} h^{2} \sup_{[a ; b]}\left|f^{\prime \prime }(x)\right|$.

Prove inequality (9) for the error of the trapezoidal rule.

Here $[a ; b]$ is divided into $n$ equal parts, $h = (b-a) / n$, $x_{i} = a+i h$, $0 \leqslant i \leqslant n$, the trapezoidal rule is $\int_{a}^{b} f(x) d x \approx h\left(\frac{1}{2}\left(f\left(x_{0}\right)+f\left(x_{n}\right)\right)+\sum_{i = 1}^{n-1} f\left(x_{i}\right)\right)$, and $\Delta$ is the difference between the integral and the right-hand side. Inequality (9): if $f$ has a piecewise continuous second derivative on $[a ; b]$, then $\left|\Delta \right| \leqslant \frac{b-a}{12} h^{2} \sup_{[a ; b]}\left|f^{\prime \prime }(x)\right|$.

Prove that Simpson's formula is an exact equality (exact) for any polynomial up to the third degree inclusive.

Prove that if an approximate integration formula

where $x_{i} = x_{0}+j h, h = 1,2$, is an exact equality for any polynomial up to degree two inclusive, then this is Simpson's formula, i.e. $p_{0} = p_{2} = h / 3, p_{1} = 4 h / 3$.

For a function $f$ having four continuous derivatives on $[a ; b]$, estimate the error of the formula obtained in problem 41, taking as known that for the error of Chebyshev's formula (22) the estimate

Using the Maclaurin series for $f(x)$, compute the integral $\int_{0}^{1} f(x) d x$ to within $10^{-3}$, if:

Compute to within $10^{-3}$: