Approximate ${\int }_3^7{x}^{2}dx$ with a MIDPOINT Riemann sum using $4$ equal subintervals.

A student writes $2\left[f\left(2\right)+f\left(4\right)\right]$ to approximate ${\int }_2^{6}f(x)dx$. Which rule produced it?

A student writes $2\left[f\left(1\right)+f\left(3\right)\right]$ to approximate ${\int }_1^{5}f(x)dx$. Which rule produced it?

A student writes $1\left[f\left(4\right)+f\left(5\right)\right]$ to approximate ${\int }_3^{5}f(x)dx$. Which rule produced it?

Approximate ${\int }_1^{5}2x^{2}dx$ with a LEFT Riemann sum using $2$ equal subintervals.

The graph shows approximating rectangles under a curve on a labeled interval. What is the total area of the rectangles?

Use the labeled graph of approximating rectangles under a curve. What is the total area of the rectangles?

Approximate ${\int }_2^{5}4x^{2}dx$ with a RIGHT Riemann sum using $3$ equal subintervals.

Approximate ${\int }_0^{4}2x^{2}dx$ with a LEFT Riemann sum using $2$ equal subintervals.

Approximate ${\int }_1^{5}2x^{2}dx$ with a MIDPOINT Riemann sum using $2$ equal subintervals.

Approximate ${\int }_2^{6}2x^{2}dx$ with a RIGHT Riemann sum using $2$ equal subintervals.

Approximate ${\int }_3^{7}2x^{2}dx$ with a RIGHT Riemann sum using $4$ equal subintervals.

Approximate ${\int }_3^{7}3x^{2}dx$ with a RIGHT Riemann sum using $2$ equal subintervals.

Approximate ${\int }_3^{5}4x^{2}dx$ with a MIDPOINT Riemann sum using $2$ equal subintervals.

Approximate ${\int }_0^{4}3x^{2}dx$ with a LEFT Riemann sum using $2$ equal subintervals.

Approximate ${\int }_3^{5}4x^{2}dx$ with a LEFT Riemann sum using $2$ equal subintervals.

Approximate ${\int }_2^{4}2x^{2}dx$ with a RIGHT Riemann sum using $2$ equal subintervals.

Approximate ${\int }_3^{5}3x^{2}dx$ with a MIDPOINT Riemann sum using $2$ equal subintervals.

Approximate ${\int }_3^{6}4x^{2}dx$ with a LEFT Riemann sum using $3$ equal subintervals.

Approximate ${\int }_0^{2}4x^{2}dx$ with a LEFT Riemann sum using $2$ equal subintervals.

Approximate ${\int }_0^{2}2x^{2}dx$ with a LEFT Riemann sum using $2$ equal subintervals.

A labeled figure shows approximating rectangles for a curve over an interval. What is the total area of the rectangles?

Approximate ${\int }_3^{5}4x^{2}dx$ with a RIGHT Riemann sum using $2$ equal subintervals.

The plotted curve has approximating rectangles drawn across the labeled interval. What is the total area of the rectangles?

Approximate ${\int }_3^{5}2x^{2}dx$ with a MIDPOINT Riemann sum using $2$ equal subintervals.

A student writes $2\left[f\left(3\right)+f\left(5\right)\right]$ to approximate ${\int }_2^{6}f(x)dx$. Which rule produced it?

A student writes $2\left[f\left(4\right)+f\left(6\right)\right]$ to approximate ${\int }_3^{7}f(x)dx$. Which rule produced it?

Approximate ${\int }_2^{4}4x^{2}dx$ with a RIGHT Riemann sum using $2$ equal subintervals.

In the figure, approximating rectangles are drawn under a curve on the shown interval. What is the total area of the rectangles?

Approximate ${\int }_2^{6}4x^{2}dx$ with a MIDPOINT Riemann sum using $4$ equal subintervals.