Applications of Integration

Apply integrals to area, accumulation, and basic net-change interpretations.

Focus points

Interpret definite integrals as signed area.

Set up area-between-curves questions carefully.

Use accumulation ideas in applied settings.

Session plan

Start with the first two questions to warm up before moving into the harder ones.

Use the calculator only after you have set up the working or estimated the answer.

Work through the page like a focused study sheet instead of jumping around.

How to use this page

A cleaner maths workflow

1. Read the prompt carefully

These pages are meant to feel consistent with the rest of the site, but the maths stays front and center.

2. Solve in order

The question list works best as a progression rather than a random set of cards.

3. Use support tools when needed

You can jump back to the calculator or home without losing the overall flow.

Practice set

6 revision prompts

Question 1

Find the area under y = 2 x + 1 from x = 0 to x = 3 .

Question 2

Find the area enclosed by y = x^{2} and y = 2 x .

Question 3OpenStax Calculus Volume 1, Sec. 5.1 Approximating Areas

Use a midpoint Riemann sum with 4 subintervals to approximate the area under f (x) = x^{2} on [0, 2] .

Question 4

The velocity of a particle is v (t) = 3 t^{2} - 4 t . Find the displacement from t = 0 to t = 3 .

Question 5

Set up an integral for the area between y = \sin x and the x -axis on [0, π] .

Question 6

If R (x) = 5 x + 2 is a rate of flow, find the accumulated quantity over [1, 4] .