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Calculus 1 : How to find trapezoidal approximation by graphing functions

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Example Question #11 : Trapezoidal Approximation

Evaluate the integral using the trapezoidal approximation:

$\int_{0}^{\pi}2e^{\cos(x)}dx$

Possible Answers:

$\pi(e^{-1}+e^1)$

$2\pi(e^{-1}+e^1)$

$\frac{\pi(e^{-1}+e^1)}{4}$

Correct answer:

$\pi(e^{-1}+e^1)$

Explanation:

To evaluate a definite integral using the trapezoidal approximation, we use the following formula:

$\int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(b)+f(a)}{2})$

Using the formula, we get

$(2)(\pi)(\frac{e^{-1}+e^1}{2})=\pi(e^{-1}+e^1)$

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Example Question #12 : Trapezoidal Approximation

Integrate using the trapezoidal approximation:

$\int_{-2}^{2}(-x^2+e^{2x})dx$

Possible Answers:

$-16+2e^4+2e^{-4}$

$-8+e^4+e^{-4}$

$2e^4-2e^{-4}$

Correct answer:

$-16+2e^4+2e^{-4}$

Explanation:

To solve the integral using the trapezoidal approximation, we must use the following formula:

$\int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(b)+f(a)}{2})$

Now, using the above formula, we can approximate the integral:

$\int_{-2}^{2}(-x^2+e^{2x})dx\approx (2+2)(\frac{-4+e^{4}-4+e^{-4}}{2})=-16+2e^4+2e^{-4}$

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Example Question #93 : Intervals

Evaluate the following integral using the trapezoidal approximation:

$\int_{a}^{b}5x^2dx$

where a and b are constants, b<a

Possible Answers:

$\frac{(a-b)(5a^2+5b^2)}{2}$

$-\frac{(a-b)(5a^2-5b^2)}{2}$

$-\frac{(a-b)(5a^2+5b^2)}{2}$

(b-a)(5a^2+5b^2)

$\frac{(b-a)(5a^2+5b^2)}{2}$

Correct answer:

$-\frac{(a-b)(5a^2+5b^2)}{2}$

Explanation:

To solve the integral, it is easiest to first rewrite it so that the larger value is the upper bound. This is done by switching the bounds and also making the integral negative:

$-\int_{b}^{a}5x^2dx$

Now, we can use the trapezoidal approximation, which states that for a definite integral:

$\int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(b)+f(a)}{2})$

Using the above formula for our integral, we get

$-(a-b)(\frac{5a^2+5b^2}{2})$

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Calculus 1 : How to find trapezoidal approximation by graphing functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #11 : Trapezoidal Approximation

Example Question #12 : Trapezoidal Approximation

Example Question #93 : Intervals

All Calculus 1 Resources

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