Properties of integrals help us simplify and evaluate definite integrals without always finding an antiderivative first. These rules include splitting sums and differences, pulling out constant multiples, reversing limits, using zero-width intervals, and adding adjacent intervals. Many problems use given integral values to solve for new integral expressions.
Notes
| \({\text{Properties of Integrals}}\) |
| \(\underline{\text{Name}}\) |
\(\underline{\text{Formula}}\) |
| \(\text{Sum & Difference}\) |
\(\displaystyle \int_{a}^{b}\left(f(x) \pm g(x)\right)\,dx \,=\, \int_{a}^{b}f(x)\,dx \pm \int_{a}^{b}g(x)\,dx\) |
| \(\text{Constant Multiple}\) |
\(\displaystyle \int_{a}^{b}\left(k \cdot f(x) \right)\,dx \,=\, k\cdot \int_{a}^{b}f(x)\,dx \) |
| \(\text{Order of Interval}\) |
\(\displaystyle \int_{a}^{b}f(x) \,dx \,=\, – \int_{b}^{a}f(x)\,dx \) |
| \(\text{Zero}\) |
\(\displaystyle \int_{a}^{a}f(x)\,dx \,=\, 0\) |
| \(\text{Adding Intervals}\) |
\(\displaystyle \int_{a}^{b}f(x)\,dx + \int_{b}^{c}f(x)\,dx \,=\, \int_{a}^{c}f(x)\,dx\) |
Practice Problems
\(\textbf{1)}\) \( \displaystyle \int_{2}^{2}f(x)\,dx \)
The answer is \(0 \)
\(\,\,\,\,\,\displaystyle \int_{a}^{a}f(x)\,dx \,=\, 0\)
\(\,\,\,\,\,\displaystyle \int_{2}^{2}f(x)\,dx\)
\(\,\,\,\,\,\text{When the lower and upper limits are the same, the integral equals }0.\)
\(\,\,\,\,\,\displaystyle \int_{a}^{a}f(x)\,dx=0\)
\(\,\,\,\,\,\displaystyle \int_{2}^{2}f(x)\,dx=0\)
\(\,\,\,\,\,\)The answer is \(0\)
\(\textbf{2)}\) \( \displaystyle \int_{2}^{9}3\,dx \)
\(3(9-2)=21 \)
\(\,\,\,\,\,\displaystyle \int_{2}^{9}3\,dx\)
\(\,\,\,\,\,\text{The integral of a constant is the constant times the interval length.}\)
\(\,\,\,\,\,\displaystyle \int_{a}^{b}c\,dx=c(b-a)\)
\(\,\,\,\,\,\displaystyle \int_{2}^{9}3\,dx=3(9-2)\)
\(\,\,\,\,\,\displaystyle 3(7)=21\)
\(\,\,\,\,\,\)The answer is \(21\)
\(\textbf{3)}\) \( \displaystyle \int_{8}^{8}f(x)\,dx \)
The answer is \(0 \)
\(\,\,\,\,\,\displaystyle \int_{8}^{8}f(x)\,dx\)
\(\,\,\,\,\,\text{When the lower and upper limits are equal, the definite integral is }0.\)
\(\,\,\,\,\,\displaystyle \int_{a}^{a}f(x)\,dx=0\)
\(\,\,\,\,\,\displaystyle \int_{8}^{8}f(x)\,dx=0\)
\(\,\,\,\,\,\)The answer is \(0\)
\(\textbf{4)}\) \( \displaystyle \int_{1}^{10}5\,dx \)
\(5(10-1)=45 \)
\(\,\,\,\,\,\displaystyle \int_{1}^{10}5\,dx\)
\(\,\,\,\,\,\text{The integral of a constant is the constant times the width of the interval.}\)
\(\,\,\,\,\,\displaystyle \int_{a}^{b}c\,dx=c(b-a)\)
\(\,\,\,\,\,\displaystyle \int_{1}^{10}5\,dx=5(10-1)\)
\(\,\,\,\,\,\displaystyle 5(9)=45\)
\(\,\,\,\,\,\)The answer is \(45\)
\(\textbf{5)}\) \(\displaystyle \int_{2}^{5}f(x)\,dx=7, \enspace \displaystyle \int_{5}^{9}f(x)\,dx=4, \enspace \displaystyle \int_{2}^{12}f(x)dx=16 \)
\( \text{Solve for } \displaystyle \int_{2}^{9}f(x)\,dx \)
The answer is \( 11 \)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle\, \int_{a}^{c}f(x)\,dx \, = \, \int_{a}^{b}f(x)\,dx + \int_{b}^{c}f(x)\,dx\)
\(\text{Apply notes to this problem.}\)
\(\,\,\,\,\,\displaystyle \int_{2}^{9} f(x)\,dx \,= \,\int_{2}^{5}f(x)\,dx+\displaystyle \int_{5}^{9}f(x)\,dx \)
\(\,\,\,\,\,\displaystyle \int_{2}^{9} f(x)\,dx \,= 7+4 \)
\(\,\,\,\,\,\displaystyle \int_{2}^{9} f(x)\,dx \,= 11 \)
\(\,\,\,\,\,\)The answer is \(11\)
\(\textbf{6)}\) \(\displaystyle \int_{2}^{5}f(x)\,dx=7, \enspace \displaystyle \int_{5}^{9}f(x)\,dx=4, \enspace \displaystyle \int_{2}^{12}f(x)\,dx=16 \)
\( \text{Solve for } \displaystyle \int_{5}^{12}f(x)\,dx \)
\(\displaystyle \int_{2}^{12}f(x)\,dx-\displaystyle \int_{2}^{5}f(x)\,dx=16-7=9 \)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle\, \int_{a}^{c}f(x)\,dx \, = \, \int_{a}^{b}f(x)\,dx + \int_{b}^{c}f(x)\,dx\)
\(\text{Apply notes to this problem.}\)
\(\,\,\,\,\,\displaystyle \int_{2}^{12}f(x)\,dx = \int_{2}^{5}f(x)\,dx + \int_{5}^{12}f(x)\,dx\)
\(\,\,\,\,\,\displaystyle 16 = 7 + \int_{5}^{12}f(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{5}^{12}f(x)\,dx = 16-7\)
\(\,\,\,\,\,\displaystyle \int_{5}^{12}f(x)\,dx = 9\)
\(\,\,\,\,\,\)The answer is \(9\)
\(\textbf{7)}\) \(\displaystyle \int_{2}^{5}f(x)\,dx=7, \enspace \displaystyle \int_{5}^{9}f(x)\,dx=4, \enspace \displaystyle \int_{2}^{12}f(x)\,dx=16 \)
\( \text{Solve for } \displaystyle \int_{9}^{5}f(x)\,dx \)
\(-\displaystyle \int_{5}^{9}f(x)\,dx=-4 \)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle \int_{b}^{a}f(x)\,dx=-\int_{a}^{b}f(x)\,dx\)
\(\text{Apply notes to this problem.}\)
\(\,\,\,\,\,\displaystyle \int_{9}^{5}f(x)\,dx=-\int_{5}^{9}f(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{9}^{5}f(x)\,dx=-4\)
\(\,\,\,\,\,\)The answer is \(-4\)
\(\textbf{8)}\) \(\displaystyle \int_{2}^{5}f(x)\,dx=7, \enspace \displaystyle \int_{5}^{9}f(x)\,dx=4, \enspace \displaystyle \int_{2}^{12}f(x)\,dx=16 \)
\( \text{Solve for } \displaystyle \int_{2}^{5}4f(x)\,dx \)
\(4\displaystyle \int_{2}^{5}f(x)\,dx=4\cdot7=28\)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle \int_{a}^{b}c\,f(x)\,dx=c\int_{a}^{b}f(x)\,dx\)
\(\text{Apply notes to this problem.}\)
\(\,\,\,\,\,\displaystyle \int_{2}^{5}4f(x)\,dx=4\int_{2}^{5}f(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{2}^{5}4f(x)\,dx=4(7)\)
\(\,\,\,\,\,\displaystyle \int_{2}^{5}4f(x)\,dx=28\)
\(\,\,\,\,\,\)The answer is \(28\)
\(\textbf{9)}\) If\( \displaystyle \int_{a}^{b}f(x)\,dx=5a-2b \), then what is \( \displaystyle \int_{a}^{b}[f(x)-3]dx? \)
\(5a-2b-3(b-a)=8a-5b \)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle \int_{a}^{b}[f(x)-g(x)]\,dx=\int_{a}^{b}f(x)\,dx-\int_{a}^{b}g(x)\,dx\)
\(\text{Apply notes to this problem.}\)
\(\,\,\,\,\,\displaystyle \int_{a}^{b} [f(x)-3]\,dx\)
\(\,\,\,\,\,\displaystyle \int_{a}^{b}f(x)\,dx – \displaystyle \int_{a}^{b}3\,dx \)
\(\,\,\,\,\,\displaystyle \int_{a}^{b}f(x)\,dx – 3 \displaystyle \int_{a}^{b}1\,dx \)
\(\,\,\,\,\,\displaystyle 5a-2b – 3(b-a) \)
\(\,\,\,\,\,\displaystyle 5a-2b – 3b +3a \)
\(\,\,\,\,\,\displaystyle 8a-5b \)
\(\,\,\,\,\,\)The answer is \(8a-5b\)
\(\textbf{10)}\) If \(\displaystyle \int_{1}^{4}f(x)\,dx=6\) and \(\displaystyle \int_{1}^{4}g(x)\,dx=10\), find \(\displaystyle \int_{1}^{4}\left[f(x)+g(x)\right]\,dx\).
The answer is \(16\)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle \int_a^b\left[f(x)+g(x)\right]\,dx=\int_a^bf(x)\,dx+\int_a^bg(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{1}^{4}\left[f(x)+g(x)\right]\,dx=6+10\)
\(\,\,\,\,\,\displaystyle \int_{1}^{4}\left[f(x)+g(x)\right]\,dx=16\)
\(\,\,\,\,\,\)The answer is \(16\)
\(\textbf{11)}\) If \(\displaystyle \int_{0}^{6}f(x)\,dx=14\) and \(\displaystyle \int_{0}^{6}g(x)\,dx=5\), find \(\displaystyle \int_{0}^{6}\left[2f(x)-3g(x)\right]\,dx\).
The answer is \(13\)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle \int_a^b\left[2f(x)-3g(x)\right]\,dx=2\int_a^bf(x)\,dx-3\int_a^bg(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{0}^{6}\left[2f(x)-3g(x)\right]\,dx=2(14)-3(5)\)
\(\,\,\,\,\,\displaystyle \int_{0}^{6}\left[2f(x)-3g(x)\right]\,dx=28-15\)
\(\,\,\,\,\,\displaystyle \int_{0}^{6}\left[2f(x)-3g(x)\right]\,dx=13\)
\(\,\,\,\,\,\)The answer is \(13\)
\(\textbf{12)}\) If \(\displaystyle \int_{3}^{8}f(x)\,dx=12\), find \(\displaystyle \int_{8}^{3}f(x)\,dx\).
The answer is \(-12\)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle \int_b^a f(x)\,dx=-\int_a^b f(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{8}^{3}f(x)\,dx=-\int_{3}^{8}f(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{8}^{3}f(x)\,dx=-12\)
\(\,\,\,\,\,\)The answer is \(-12\)
\(\textbf{13)}\) If \(\displaystyle \int_{0}^{3}f(x)\,dx=9\) and \(\displaystyle \int_{3}^{7}f(x)\,dx=11\), find \(\displaystyle \int_{0}^{7}f(x)\,dx\).
The answer is \(20\)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle \int_a^c f(x)\,dx=\int_a^b f(x)\,dx+\int_b^c f(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{0}^{7}f(x)\,dx=\int_{0}^{3}f(x)\,dx+\int_{3}^{7}f(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{0}^{7}f(x)\,dx=9+11\)
\(\,\,\,\,\,\displaystyle \int_{0}^{7}f(x)\,dx=20\)
\(\,\,\,\,\,\)The answer is \(20\)
\(\textbf{14)}\) If \(\displaystyle \int_{-2}^{5}f(x)\,dx=18\) and \(\displaystyle \int_{-2}^{1}f(x)\,dx=7\), find \(\displaystyle \int_{1}^{5}f(x)\,dx\).
The answer is \(11\)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle \int_{-2}^{5}f(x)\,dx=\int_{-2}^{1}f(x)\,dx+\int_{1}^{5}f(x)\,dx\)
\(\,\,\,\,\,\displaystyle 18=7+\int_{1}^{5}f(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{1}^{5}f(x)\,dx=18-7\)
\(\,\,\,\,\,\displaystyle \int_{1}^{5}f(x)\,dx=11\)
\(\,\,\,\,\,\)The answer is \(11\)
\(\textbf{15)}\) If \(\displaystyle \int_{4}^{10}f(x)\,dx=15\), find \(\displaystyle \int_{4}^{10}6f(x)\,dx\).
The answer is \(90\)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle \int_a^b kf(x)\,dx=k\int_a^b f(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{4}^{10}6f(x)\,dx=6\int_{4}^{10}f(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{4}^{10}6f(x)\,dx=6(15)\)
\(\,\,\,\,\,\displaystyle \int_{4}^{10}6f(x)\,dx=90\)
\(\,\,\,\,\,\)The answer is \(90\)
\(\textbf{16)}\) If \(\displaystyle \int_{1}^{6}f(x)\,dx=-3\), find \(\displaystyle \int_{1}^{6}\left[f(x)+4\right]\,dx\).
The answer is \(17\)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle \int_a^b\left[f(x)+4\right]\,dx=\int_a^b f(x)\,dx+\int_a^b4\,dx\)
\(\,\,\,\,\,\displaystyle \int_{1}^{6}\left[f(x)+4\right]\,dx=-3+4(6-1)\)
\(\,\,\,\,\,\displaystyle \int_{1}^{6}\left[f(x)+4\right]\,dx=-3+20\)
\(\,\,\,\,\,\displaystyle \int_{1}^{6}\left[f(x)+4\right]\,dx=17\)
\(\,\,\,\,\,\)The answer is \(17\)
\(\textbf{17)}\) If \(\displaystyle \int_{0}^{2}f(x)\,dx=5\), \(\displaystyle \int_{0}^{2}g(x)\,dx=-1\), find \(\displaystyle \int_{0}^{2}\left[3f(x)+2g(x)-4\right]\,dx\).
The answer is \(5\)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle \int_a^b\left[3f(x)+2g(x)-4\right]\,dx=3\int_a^bf(x)\,dx+2\int_a^bg(x)\,dx-\int_a^b4\,dx\)
\(\,\,\,\,\,\displaystyle \int_{0}^{2}\left[3f(x)+2g(x)-4\right]\,dx=3(5)+2(-1)-4(2-0)\)
\(\,\,\,\,\,\displaystyle \int_{0}^{2}\left[3f(x)+2g(x)-4\right]\,dx=15-2-8\)
\(\,\,\,\,\,\displaystyle \int_{0}^{2}\left[3f(x)+2g(x)-4\right]\,dx=5\)
\(\,\,\,\,\,\)The answer is \(5\)
\(\textbf{18)}\) If \(\displaystyle \int_{-1}^{4}f(x)\,dx=13\), find \(\displaystyle \int_{4}^{-1}2f(x)\,dx\).
The answer is \(-26\)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle \int_b^a2f(x)\,dx=-\int_a^b2f(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{4}^{-1}2f(x)\,dx=-2\int_{-1}^{4}f(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{4}^{-1}2f(x)\,dx=-2(13)\)
\(\,\,\,\,\,\displaystyle \int_{4}^{-1}2f(x)\,dx=-26\)
\(\,\,\,\,\,\)The answer is \(-26\)
\(\textbf{19)}\) If \(\displaystyle \int_{0}^{5}f(x)\,dx=8\) and \(\displaystyle \int_{5}^{9}f(x)\,dx=-6\), find \(\displaystyle \int_{9}^{0}f(x)\,dx\).
The answer is \(-2\)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle \int_{0}^{9}f(x)\,dx=\int_{0}^{5}f(x)\,dx+\int_{5}^{9}f(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{0}^{9}f(x)\,dx=8+(-6)=2\)
\(\,\,\,\,\,\displaystyle \int_{9}^{0}f(x)\,dx=-\int_{0}^{9}f(x)\,dx\)
\(\,\,\,\,\,\displaystyle \int_{9}^{0}f(x)\,dx=-2\)
\(\,\,\,\,\,\)The answer is \(-2\)
\(\textbf{20)}\) If \(\displaystyle \int_{a}^{b}f(x)\,dx=12\), \(\displaystyle \int_{a}^{b}g(x)\,dx=7\), and \(b-a=5\), find \(\displaystyle \int_{a}^{b}\left[4f(x)-g(x)+2\right]\,dx\).
The answer is \(51\)
\(\text{Notes}\)
\(\,\,\,\,\,\displaystyle \int_a^b\left[4f(x)-g(x)+2\right]\,dx=4\int_a^bf(x)\,dx-\int_a^bg(x)\,dx+\int_a^b2\,dx\)
\(\,\,\,\,\,\displaystyle \int_{a}^{b}\left[4f(x)-g(x)+2\right]\,dx=4(12)-7+2(b-a)\)
\(\,\,\,\,\,\displaystyle \int_{a}^{b}\left[4f(x)-g(x)+2\right]\,dx=48-7+2(5)\)
\(\,\,\,\,\,\displaystyle \int_{a}^{b}\left[4f(x)-g(x)+2\right]\,dx=51\)
\(\,\,\,\,\,\)The answer is \(51\)
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