Step-by-Step Examples

Example

Approximate the area under the curve f(x)=x2f(x) = x^2 on the interval [0,2][0, 2] using a right Riemann sum with n=4n = 4 subintervals.

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Example

Use the Trapezoidal Rule with n=4n = 4 to approximate the integral:
∫131x dx \int_1^3 \frac{1}{x} \, dx

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Example

Evaluate the integral geometrically by recognizing the curve:
∫−339−x2 dx \int_{-3}^3 \sqrt{9 - x^2} \, dx

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Example

Evaluate the definite integral using the Fundamental Theorem of Calculus:
∫13(2x+1) dx \int_1^3 (2x + 1) \, dx

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Example

Evaluate the definite integral involving an odd function over a symmetric interval:
∫−ππsin⁡(x)cos⁡2(x) dx \int_{-\pi}^{\pi} \sin(x) \cos^2(x) \, dx

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Example

Evaluate the definite integral of a trigonometric function:
∫0πsin⁡x dx \int_0^{\pi} \sin x \, dx

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Example

Evaluate using substitution:
∫012x(x2+1)3 dx \int_0^1 2x(x^2 + 1)^3 \, dx

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Mean Value Theorem for Definite Integrals Examples

Example

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

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Improper Integrals Examples

Example

Evaluate the improper integral with an infinite interval:
∫1∞1x2 dx \int_1^{\infty} \frac{1}{x^2} \, dx

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Example

Evaluate the improper integral with an infinite discontinuity:
∫0313−x dx \int_0^3 \frac{1}{\sqrt{3-x}} \, dx

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Example

Determine if the improper integral converges or diverges:
∫1∞1x dx \int_1^{\infty} \frac{1}{x} \, dx

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Key Takeaways
  • The definite integral calculates the exact signed area bounded by a curve and the x-axis over a specified interval.
  • Riemann sums provide the theoretical foundation, approximating area through discrete rectangles. The limit as rectangles become infinitely thin yields the exact integral.
  • The Fundamental Theorem of Calculus bridges differential and integral calculus. It enables the evaluation of definite integrals simply by finding an antiderivative and subtracting its values at the interval boundaries (F(b)−F(a)F(b) - F(a)).
  • When applying u-substitution to definite integrals, it is crucial to convert the limits of integration from xx-values to uu-values to streamline the evaluation process.
  • Improper integrals with infinite limits or discontinuities must be evaluated using limits to determine if they converge to a finite value or diverge.
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