Practice Riemann Sums in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A method of approximating the definite integral ∫abf(x) dx\int_a^b f(x)\,dx by dividing the interval [a,b][a, b] into subintervals and summing the areas of rectangles (or trapezoids) whose heights are determined by the function.

Imagine filling the area under a curve with thin rectangles. The more rectangles you use, the better the approximation. In the limit of infinitely many infinitely thin rectangles, you get the exact areaβ€”which is the definite integral.

Showing a random 20 of 50 problems.

Example 1

easy
What does a Riemann sum approximate as nβ†’βˆžn\to\infty?

Example 2

medium
Estimate ∫03x2 dx\int_0^3 x^2\,dx using a left Riemann sum with n=3n=3.

Example 3

easy
As nβ†’βˆžn \to \infty, Ξ”x=(bβˆ’a)/n\Delta x = (b-a)/n approaches what value?

Example 4

medium
Same table as the previous question. Find the right Riemann sum for ∫04f(x) dx\int_0^4 f(x)\,dx with n=4n=4.

Example 5

easy
For ∫06f(x) dx\int_0^6 f(x)\,dx with n=6n=6, list the right endpoints.

Example 6

challenge
For ∫01(3x2) dx\int_0^1 (3x^2)\,dx, show the limit of right Riemann sums equals 1.

Example 7

medium
Estimate ∫04x2 dx\int_0^4 x^2\,dx using a right Riemann sum with n=4n=4.

Example 8

hard
Express ∫02x3 dx\int_0^2 x^3\,dx as the limit of a right Riemann sum, then evaluate.

Example 9

easy
Use a midpoint sum with n=2n=2 to approximate ∫04(x+1) dx\int_0^4 (x+1)\,dx.

Example 10

medium
Approximate ∫04x2 dx\int_0^4 x^2\,dx using a midpoint sum with n=4n=4.

Example 11

medium
For an increasing function on [a,b][a,b], which Riemann sum is an upper bound: left, right, or midpoint?

Example 12

medium
Estimate ∫04x2 dx\int_0^4 x^2\,dx using the trapezoidal rule with n=4n=4.

Example 13

easy
Estimate ∫04(x+1) dx\int_0^4 (x+1)\,dx with a left sum, n=2n=2.

Example 14

medium
For ∫04x dx\int_0^4 \sqrt{x}\,dx, n=4n=4, find the midpoint sum (3 decimal places).

Example 15

medium
Estimate ∫131x dx\int_1^3 \frac{1}{x}\,dx using a midpoint sum with n=2n=2.

Example 16

easy
Approximate ∫02x2 dx\int_0^2 x^2\,dx using a left Riemann sum with n=4n = 4 equal subintervals.

Example 17

hard
For ∫01x2 dx\int_0^1 x^2\,dx, evaluate the right Riemann sum as a function of nn and take the limit.

Example 18

medium
Approximate ∫131x dx\int_1^3 \frac{1}{x}\,dx using a right Riemann sum with n=4n = 4 subintervals and classify the estimate.

Example 19

easy
For ∫04f(x) dx\int_0^4 f(x)\,dx with n=4n=4 subintervals, what is Ξ”x\Delta x?

Example 20

challenge
Evaluate ∫01x2 dx\int_0^1 x^2\,dx from the definition using the limit of right Riemann sums.
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Related Concepts

IntegralDefinite Integral