Properties of Integrals

Andymath.com features free videos, notes, and practice problems with answers! Printable pages make math easy. Are you ready to be a mathmagician?

Notes

\({\text{Properties of Integrals}}\)
\(\underline{\text{Name}}\) \(\underline{\text{Formula}}\)
\(\text{Sum & Difference}\)
\(\displaystyle \int_{a}^{b}\left(f(x) \pm g(x)\right)\,dx \,=\, \int_{a}^{b}f(x)\,dx \pm \int_{a}^{b}g(x)\,dx\)
\(\text{Constant Multiple}\)
\(\displaystyle \int_{a}^{b}\left(k \cdot f(x) \right)\,dx \,=\, k\cdot \int_{a}^{b}f(x)\,dx \)
\(\text{Order of Interval}\)
\(\displaystyle \int_{a}^{b}f(x) \,dx \,=\, – \int_{b}^{a}f(x)\,dx \)
\(\text{Zero}\)
\(\displaystyle \int_{a}^{a}f(x)\,dx \,=\, 0\)
\(\text{Adding Intervals}\)
\(\displaystyle \int_{a}^{b}f(x)\,dx + \int_{b}^{c}f(x)\,dx \,=\, \int_{a}^{c}f(x)\,dx\)


Questions & Solutions

\(\textbf{1)}\) \( \displaystyle \int_{2}^{2}f(x)\,dx \)




\(\textbf{2)}\) \( \displaystyle \int_{2}^{9}3\,dx \)




\(\textbf{3)}\) \( \displaystyle \int_{8}^{8}f(x)\,dx \)




\(\textbf{4)}\) \( \displaystyle \int_{1}^{10}5\,dx \)




\(\textbf{5)}\) \(\displaystyle \int_{2}^{5}f(x)\,dx=7, \enspace \displaystyle \int_{5}^{9}f(x)\,dx=4, \enspace \displaystyle \int_{2}^{12}f(x)dx=16 \)
\( \text{Solve for } \displaystyle \int_{2}^{9}f(x)\,dx \)




\(\textbf{6)}\) \(\displaystyle \int_{2}^{5}f(x)\,dx=7, \enspace \displaystyle \int_{5}^{9}f(x)\,dx=4, \enspace \displaystyle \int_{2}^{12}f(x)\,dx=16 \)
\( \text{Solve for } \displaystyle \int_{5}^{12}f(x)\,dx \)




\(\textbf{7)}\) \(\displaystyle \int_{2}^{5}f(x)\,dx=7, \enspace \displaystyle \int_{5}^{9}f(x)\,dx=4, \enspace \displaystyle \int_{2}^{12}f(x)\,dx=16 \)
\( \text{Solve for } \displaystyle \int_{9}^{5}f(x)\,dx \)




\(\textbf{8)}\) \(\displaystyle \int_{2}^{5}f(x)\,dx=7, \enspace \displaystyle \int_{5}^{9}f(x)\,dx=4, \enspace \displaystyle \int_{2}^{12}f(x)\,dx=16 \)
\( \text{Solve for } \displaystyle \int_{2}^{5}4f(x)\,dx \)




\(\textbf{9)}\) If\( \displaystyle \int_{a}^{b}f(x)\,dx=5a-2b \), then what is \( \displaystyle \int_{a}^{b}[f(x)-3]dx? \)



See Related Pages\(\)

\(\bullet\text{ Trapezoidal Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{b-a}{2n}\left[f(a)+2f(x_1)+2f(x_2)+…+2fx_{n-1}+f(b)\right]…\)
\(\bullet\text{ Properties of Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}cf(x) \, dx=c\displaystyle \int_{a}^{b}f(x) \,dx…\)
\(\bullet\text{ Indefinite Integrals- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int x^n \, dx = \displaystyle \frac{x^{n+1}}{n+1}+C…\)
\(\bullet\text{ Indefinite Integrals- Trig Functions}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int \cos{x} \, dx=\sin{x}+C…\)
\(\bullet\text{ Definite Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{5}^{7} x^3 \, dx…\)
\(\bullet\text{ Integration by Substitution}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int (x^2+3)^3(2x) \,dx…\)
\(\bullet\text{ Area of Region Between Two Curves}\)
\(\,\,\,\,\,\,\,\,A=\displaystyle \int_{a}^{b}\left[f(x)-g(x)\right]\,dx…\)
\(\bullet\text{ Arc Length}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}\sqrt{1+\left[f'(x)\right]^2} \,dx…\)
\(\bullet\text{ Average Function Value}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{1}{b-a} \int_{a}^{b}f(x) \,dx\)
\(\bullet\text{ Volume by Cross Sections}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Disk Method}\)
\(\,\,\,\,\,\,\,\,V=\displaystyle \int_{a}^{b}\left[f(x)\right]^2\,dx…\)
\(\bullet\text{ Cylindrical Shells}\)
\(\,\,\,\,\,\,\,\,V=2 \pi \displaystyle \int_{a}^{b} y f(y) \, dy…\)
\(\bullet\text{ Andymath Homepage}\)

Thumbnail of Andymath Homepage

About Andymath.com

Andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. If you have any requests for additional content, please contact Andy at tutoring@andymath.com. He will promptly add the content.

Topics cover Elementary Math, Middle School, Algebra, Geometry, Algebra 2/Pre-calculus/Trig, Calculus and Probability/Statistics. In the future, I hope to add Physics and Linear Algebra content.

Visit me on Youtube, Tiktok, Instagram and Facebook. Andymath content has a unique approach to presenting mathematics. The clear explanations, strong visuals mixed with dry humor regularly get millions of views. We are open to collaborations of all types, please contact Andy at tutoring@andymath.com for all enquiries. To offer financial support, visit my Patreon page. Let’s help students understand the math way of thinking!

Thank you for visiting. How exciting!