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Back to Integral Calculus

The Definite Integral

The limit of Riemann sums is a signed area: above the axis counts positive, below counts negative, and running the interval backwards flips it all.

Step 1 of 20

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f(x) = 0.75(x − 1)(x − 3) dips below the x-axis between 1 and 3. What should "the area under it" from 0 to 4 mean, when part of it is under the axis?

\int_{0}^{4} \tfrac{3}{4}(x - 1)(x - 3)\,dx = F(4) - F(0) = 1

\textcolor{#facc15}{2} + (\textcolor{#fb7185}{-1}) = 1, \qquad \text{total area ignoring signs} = 3

F is an antiderivative of f — here F(x) = 0.25x³ − 1.5x² + 2.25x — and why F(b) − F(a) gives the area is the Fundamental Theorem, next in this section.

The Definite Integral

The limit of Riemann sums is a signed area: above the axis counts positive, below counts negative, and running the interval backwards flips it all.

Step 1 of 20

Speed

f(x) = 0.75(x − 1)(x − 3) dips below the x-axis between 1 and 3. What should "the area under it" from 0 to 4 mean, when part of it is under the axis?

\int_{0}^{4} \tfrac{3}{4}(x - 1)(x - 3)\,dx = F(4) - F(0) = 1

\textcolor{#facc15}{2} + (\textcolor{#fb7185}{-1}) = 1, \qquad \text{total area ignoring signs} = 3

F is an antiderivative of f — here F(x) = 0.25x³ − 1.5x² + 2.25x — and why F(b) − F(a) gives the area is the Fundamental Theorem, next in this section.

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The Definite Integral

See also

The Definite Integral

See also