Slice the region under a curve into thin rectangles and add them. Thinner slices give a better total.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The integral is the limit
The definite integral is what that sum approaches as the slices become infinitely thin.
It is signed area
Area below the axis counts as negative. An integral can be zero over a region with plenty of area.
The Fundamental Theorem
To integrate, find an antiderivative and subtract its values at the two ends. Differentiation and integration undo each other.
The integral accumulates
The integral of a rate over an interval gives the total accumulated. Integrating a flow rate gives volume; integrating velocity gives displacement. That reading is what most free-response questions are testing.
Riemann sums define it
Slice into thin rectangles, sum the areas, and take the limit as the widths shrink. Left, right and midpoint sums all converge to the same value for a continuous function, at different speeds.
Signed area
Regions below the axis contribute negatively. Integrating velocity gives displacement; total distance requires integrating the absolute value, or splitting at the sign changes. The exam tests this distinction directly.
Units come from the product
Integrating litres per minute with respect to minutes gives litres. Working out the units of an integral from the integrand and the variable is a reliable way to interpret it in context.
Step 2: Try It Yourself
Shade between the two points and read the accumulated area.
- Point(0, 0)
- Second point(3, 9)
- Slope between them3
Slide the second point towards the first. The slope between them approaches the slope of the curve at that point.
Step 3: Watch an Example
One step at a time.
Watch Nadia Apply the Fundamental Theorem
Nadia integrates 2x from 1 to 3.
- Step 1
An antiderivative of 2x is x².
Step 4: Your Turn
Practice makes it stick.
The Antiderivative
Problem 1 of 2
Integrate 2x from 0 to 4. What is the value?
Signed Area
Problem 2 of 2
Integrate x from −2 to 2. What is the value?
Accumulate
1 of 8
Integrate 2x from 0 to 5.
2 of 8
Integrate 3 from 0 to 4.
3 of 8
Integrate 3x² from 0 to 2.
4 of 8
Integrate 2x from 3 to 3.
5 of 8
Select every true statement about the definite integral.
6 of 8
Integrate 2x from 4 to 0. What is the value?
7 of 8
An antiderivative of 4x³ is x⁴. Integrate from 0 to 2.
8 of 8
Integrate 5 from 2 to 7.
Step 5: Quick Check
Show what you know.
Question 1 of 1
Integrate 2x from 0 to 6.
What You Learned
- A definite integral is the limit of a sum of infinitely thin slices.
- It measures signed area, so regions below the axis subtract.
- The Fundamental Theorem evaluates it by subtracting an antiderivative at the two ends.