Calculus Riemann sum Questions

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Calculus Riemann sum Questions

1.

Which of the following sums does not equal the others? (4 points)


2.

Estimate the area under the curve f(x) = x2 from x = 1 to x = 5 by using four inscribed (under the curve) rectangles. Answer to the nearest integer. (4 points)




3.

List x1, x2, x3, x4 where xi is the right endpoint of the four equal intervals used to estimate the area under the curve of f(x) between x = 3 and x = 5. (4 points)



4.

Write the summation to estimate the area under the curve y = 2x2 + 1 from x = 0 to x = 4 using 4 rectangles and left endpoints. (4 points)



5.

If the area under the curve of f(x) = x2 + 2 from x = 1 to x = 6 is estimated using five approximating rectangles and right endpoints, will the estimate be an underestimate or overestimate? (4 points)

1.

The Riemann sum, the limit as the maximum of delta x sub i goes to infinity of the summation from i equals 1 to n of f of the quantity x star sub i times delta x sub i , is equivalent to the limit as n goes to infinity of the summation from i equals 1 to n of f of the quantity a plus i times delta x, times delta x with delta x equals the quotient of the quantity b minus a and n .

Write the integral that produces the same value as the limit as n goes to infinity of the summation from i equals 1 to n of the product of the quantity squared of 3 plus 5 times i over n and 5 over n . (4 points)


2.

Write the Riemann sum to find the area under the graph of the function f(x) = x3 from x = 2 to x = 5. (4 points)



3.

Use your calculator to evaluate the limit from x equals 1 to 3 of e raised to the x squared power, dx . Give your answer to the nearest integer. (4 points)




4.

Use geometry to evaluate the integral from negative 2 to 2 of the quantity 2 minus the absolute value of x, dx . (4 points)



5.

Use geometry to evaluate the integral from negative 2 to 2 of the square root of the quantity 4 minus x squared, dx . (4 points)

1.

Given that the antiderivative of f of x equals 1 divided by x is F(x) = Ln|x| + C, evaluate the integral from 1 to 2 of the 1 divided by x, dx . (4 points)


2.

Evaluate the integral from negative 3 to 0 of the absolute value of the quantity x plus 2, dx . (4 points)



3.

Given G of x equals the integral from 1 to x of the natural logarithm of the quantity 2 times t plus 1, dt , find G ‘(x). (4 points)



4.

Find the derivative with respect to x of the integral from 1 to x squared of the natural logarithm of t, dt . (4 points)



5.

Determine the interval on which f(x) = ln(x) is integrable. (4 points)






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