Riemann and Darboux Sum Examples

Example

Approximate

∫02x2 dx\int_0^2 x^2\,dx

with a right Riemann sum using n=4n=4 equal subintervals.

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Example

For f(x)=x2f(x)=x^2 on [0,2][0,2] with the partition P={0,1,2}P=\{0,1,2\}, compute the lower and upper Darboux sums.

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Numerical Integration Example

Example

Use the composite Trapezoidal Rule with n=4n=4 to approximate

∫131x dx.\int_1^3\frac{1}{x}\,dx.

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Geometry and the Fundamental Theorem

Example

Evaluate geometrically

∫−339−x2 dx.\int_{-3}^{3}\sqrt{9-x^2}\,dx.

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Example

Evaluate using FTC:

∫13(2x+1) dx.\int_1^3(2x+1)\,dx.

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Signed Accumulation and Symmetry

Example

Evaluate

∫−ππsin⁡xcos⁡2x dx.\int_{-\pi}^{\pi}\sin x\cos^2x\,dx.

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Example

Evaluate

∫0πsin⁡x dx.\int_0^{\pi}\sin x\,dx.

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Definite Integral by Substitution

Example

Evaluate

∫012x(x2+1)3 dx.\int_0^1 2x(x^2+1)^3\,dx.

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Average Value Example

Example

Find the average value of f(x)=x2f(x)=x^2 on [0,3][0,3] and a point cc guaranteed by the Mean Value Theorem for Integrals.

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Improper Integral Examples

Example

Evaluate

∫1∞1x2 dx.\int_1^{\infty}\frac{1}{x^2}\,dx.

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Example

Evaluate

∫0313−x dx.\int_0^3\frac{1}{\sqrt{3-x}}\,dx.

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Example

Determine whether

∫1∞1x dx\int_1^{\infty}\frac1x\,dx

converges.

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Key Takeaways
  • Riemann and numerical sums are approximations unless a separate exactness argument applies.
  • Darboux sums use infima and suprema; these bounds need not be attained for a general bounded function.
  • A definite integral measures signed accumulation. It equals ordinary geometric area only when the integrand is nonnegative or after absolute values are handled appropriately.
  • Carry exact fractions or symbolic values as long as practical; round only when reporting a numerical approximation.
  • Improper integrals are evaluated through the defining one-sided or infinite limits, never by substituting infinity as a number.