Riemann Sums Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy

Use a midpoint sum with

n=2

to approximate

\int_0^4 (x+1)\,dx

Solution

1
$\Delta x = 2$ . Midpoints: $x = 1, 3$ .
2
$M_2 = 2(f(1)+f(3)) = 2(2+4) = 12$ .

Answer

M_2 = 12

For a linear function, the midpoint rule is exact regardless of

n

. The exact integral is also 12.

About Riemann Sums

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

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More Riemann Sums Examples

Example 1 easy

Approximate [formula] using a left Riemann sum with [formula] equal subintervals.

Example 2 medium

Approximate [formula] using a right Riemann sum with [formula] subintervals and classify the estimat

Example 4 hard

Write the right Riemann sum for [formula] with [formula] subintervals as a sigma expression and eval

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Related Concepts

Integral Definite Integral