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Calculus 1 : Trapezoidal Approximation

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Example Questions

Example Question #1 : Trapezoidal Approximation

Use the trapezoidal approximation to find the area under the curve using the graph with four partitions.

Graph1

Possible Answers:

$A\approx 29.6$

$A \approx 38.4$

$A\approx 19.2$

$A \approx 24.4$

Correct answer:

$A\approx 29.6$

Explanation:

The trapezoid rule states that

$\\ A\approx \frac{\Delta x}{2}[f(x_0)+2\sum_{i=1}^{n-1}f(x_i)+f(x_n)]\\ \Delta x=\frac{b-a}{n}, x_i=a+i\Delta x$ .

Therefore, using our graph, we have:

$\\ \Delta x=\frac{8}{4}=2\\ \\ x_i=0,2,4,6,8$

We find the function values at the sample point:

f(x_i)=6,4,4,2.4,2.8

Then we substitute the appropriate values into the trapezoid rule approximation:

$A\approx\frac{2}{2}[6+2(4+4+2.4)+2.8]=29.6$

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Example Question #2 : Trapezoidal Approximation

Is the following function increasing or decreasing at the point ?

h(x)=-x^5+5x^4+2x

Possible Answers:

h(x) is decreasing at , because the second derivative is negative.

h(x) is increasing at , because the second derivative is positive.

h(x) is decreasing at , because the first derivative is negative.

h(x) is increasing at , because the first derivative is positive.

Correct answer:

h(x) is decreasing at , because the first derivative is negative.

Explanation:

Is the following function increasing or decreasing at the point ?

h(x)=-x^5+5x^4+2x

Increasing and decreasing intervals can be found via the first derivative. Since derivatives measure rates of change, the sign of the derivative at a given point can tell you whether a function is increasing or decreasing.

Begin by taking the derivative of our function:

h(x)=-x^5+5x^4+2x

Becomes:

h'(x)=-5x^4+20x^3+2

Next, find h'(-6) and look at the sign.

h'(x)=-5(-6)^4+20(-6)^3+2=-43198

So, our first derivative is very negative at the given point. This means that h(x) is decreasing.

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Example Question #1 : How To Find Trapezoidal Approximation By Graphing Functions

Use the trapezoidal approximation to approximate the following integral:

$\int_{0}^{5}(x^2+2x+1)dx$

Possible Answers:

180

Correct answer:

Explanation:

The trapezoidal approximation of a definite integral is given by the following formula:

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

Using the above formula, we get

$5(\frac{36}{2})=90$

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Example Question #2 : How To Find Trapezoidal Approximation By Graphing Functions

Use the trapezoidal approximation to evaluate the following integral:

$\int_{1}^{3}(x^3+x^2+6x)dx$

Possible Answers:

Correct answer:

Explanation:

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

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

Using the above formula, we get

$2(\frac{27+9+10-1-1-6}{2})=46$

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Example Question #5 : Trapezoidal Approximation

Use the trapezoidal approximation to evaluate the following integral:

$\int_{0}^{\pi} 3\cos(x)dx$

Possible Answers:

$-3\pi$

$\frac{3\pi}{2}$

$6\pi$

$3\pi$

Correct answer:

$-3\pi$

Explanation:

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

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

Using the above formula, we get

$\pi(\frac{-3-3}{2})=-3\pi$

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Example Question #6 : Trapezoidal Approximation

Evaluate the following integral using the trapezoidal approximation:

$\int_{0}^{\2\pi}\frac{e^{2x}}{5\cos(x)}dx$

Possible Answers:

$2\pi (\frac{e^{4\pi}+1}{5})$

$2\pi (e^{4\pi}-1)$

$\pi (\frac{e^{4\pi}-1}{5})$

$2\pi (\frac{e^{4\pi}-1}{5})$

$2\pi (\frac{e^{4\pi}-1}{10})$

Correct answer:

$2\pi (\frac{e^{4\pi}-1}{10})$

Explanation:

To evaluate the integral using the trapezoidal rule, we must use the formula

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

Using the above formula, we get the following:

$2\pi(\frac{e^{4\pi}-1}{10})$

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Example Question #7 : Trapezoidal Approximation

Evaluate the integral using the trapezoidal approximation:

$\int_{5}^{10}( x^2+e^{x-5})dx$

Possible Answers:

$\frac{5}{2}(75+e^5)$

5(126+e^5)

$\frac{5}{2}(126-e^5)$

$\frac{5}{2}(126+e^5)$

Correct answer:

$\frac{5}{2}(126+e^5)$

Explanation:

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

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

So, using the above formula, we get

$\frac{(5)(100+e^5+26)}{2}$

which simplifies to

$\frac{5}{2}(126+e^5)$

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Example Question #8 : Trapezoidal Approximation

Evaluate the integral using the trapezoidal approximation:

$\int_{2}^{4}\frac{ t^3-t^2}{t}dt$

Possible Answers:

Correct answer:

Explanation:

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

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

Using the above formula, we get

$\frac{2(\frac{48}{4}+\frac{4}{2})}{2}=14$

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Example Question #3 : How To Find Trapezoidal Approximation By Graphing Functions

Evaluate the integral using the trapezoidal approximation:

$\int_{-2}^{2}\frac{x^2+e^{x-2}}{3x}dx$

Possible Answers:

$\frac{1+e^{-4}}{3}$

$\frac{1-e^{-4}}{3}$

$\frac{1-e^{-4}}{6}$

$\frac{9+e^{-4}}{3}$

Correct answer:

$\frac{1-e^{-4}}{3}$

Explanation:

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

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

Using the formula, we get

$\frac{4(\frac{5}{6}+\frac{4+e^{-4}}{-6})}{2}=\frac{1-e^{-4}}{3}$

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Example Question #10 : Trapezoidal Approximation

Evaluate the integral using the trapezoidal approximation:

$\int_{0}^{10} (\frac{x^2e^{10-x}}{5}+5x)dx$

Possible Answers:

350

700

250

150

Correct answer:

350

Explanation:

To evaluate the definite integral using the trapezoidal approximation, the following formula is used:

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

Using the above formula, we get

$\frac{10[(\frac{100}{5}+50)+(0)]}{2}=350$ .

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Calculus 1 : Trapezoidal Approximation

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1 : Trapezoidal Approximation

Example Question #2 : Trapezoidal Approximation

Example Question #1 : How To Find Trapezoidal Approximation By Graphing Functions

Example Question #2 : How To Find Trapezoidal Approximation By Graphing Functions

Example Question #5 : Trapezoidal Approximation

Example Question #6 : Trapezoidal Approximation

Example Question #7 : Trapezoidal Approximation

Example Question #8 : Trapezoidal Approximation

Example Question #3 : How To Find Trapezoidal Approximation By Graphing Functions

Example Question #10 : Trapezoidal Approximation

All Calculus 1 Resources

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