Definite integrals: F(b) − F(a)

For a definite integral, find an antiderivative $F(x)$, then subtract its value at the lower limit $a$ from its value at the upper limit $b$.

$C$ cancels in the subtraction, so it is not needed. Sign mistakes are common, so write $F(b)$ and $F(a)$ separately before subtracting.

Examples

1. 01What is this definite integral?$\displaystyle \int_{1}^{2} 2x \, dx$AnswerHide

   Answer$\displaystyle 3$

   Steps

   1. $\displaystyle \int_{1}^{2} 2x \, dx$$\displaystyle = \left[x^{2}\right]_{1}^{2}$$\displaystyle = 2^{2} - 1^{2}$$\displaystyle = 3$
2. 02What is this definite integral?$\displaystyle \int_{-1}^{1} 3x^2 \, dx$AnswerHide

   Answer$\displaystyle 2$

   Steps

   1. $\displaystyle \int_{-1}^{1} 3x^2 \, dx$$\displaystyle = \left[x^{3}\right]_{-1}^{1}$$\displaystyle = 1^{3} - (-1)^{3}$$\displaystyle = 2$
3. 03What is this definite integral?$\displaystyle \int_{-1}^{3} 2x \, dx$AnswerHide

   Answer$\displaystyle 8$

   Steps

   1. $\displaystyle \int_{-1}^{3} 2x \, dx$$\displaystyle = \left[x^{2}\right]_{-1}^{3}$$\displaystyle = 3^{2} - (-1)^{2}$$\displaystyle = 8$
4. 04What is this definite integral?$\displaystyle \int_{-3}^{-1} 2x \, dx$AnswerHide

   Answer$\displaystyle -8$

   Steps

   1. $\displaystyle \int_{-3}^{-1} 2x \, dx$$\displaystyle = \left[x^{2}\right]_{-3}^{-1}$$\displaystyle = (-1)^{2} - (-3)^{2}$$\displaystyle = -8$
5. 05What is this definite integral?$\displaystyle \int_{0}^{1} 3x^2 \, dx$AnswerHide

   Answer$\displaystyle 1$

   Steps

   1. $\displaystyle \int_{0}^{1} 3x^2 \, dx$$\displaystyle = \left[x^{3}\right]_{0}^{1}$$\displaystyle = 1^{3} - 0^{3}$$\displaystyle = 1$
6. 06What is this definite integral?$\displaystyle \int_{-1}^{0} 3x^2 \, dx$AnswerHide

   Answer$\displaystyle 1$

   Steps

   1. $\displaystyle \int_{-1}^{0} 3x^2 \, dx$$\displaystyle = \left[x^{3}\right]_{-1}^{0}$$\displaystyle = 0^{3} - (-1)^{3}$$\displaystyle = 1$

FAQ