Lesson
Practice Problems & Videos
\(\textbf{1)}\) \(\displaystyle\int_{2}^{6}5 \, dx\)
\(\textbf{2)}\) \(\displaystyle\int_{-3}^{3}x \, dx\)
\(\textbf{3)}\) \(\displaystyle\int_{5}^{7}x^3 \, dx\)
\(\textbf{4)}\) \(\displaystyle\int_{1}^{2}\left(5x^2+4x\right) \, dx\)
\(\textbf{5)}\) \(\displaystyle\int_{0}^{1}\left(2x^5-3x^2\right) \, dx\)
\(\textbf{6)}\) \(\displaystyle\int_{1}^{2}10x^9 \, dx\)
\(\textbf{7)}\) \(\displaystyle\int_{1}^{3}2x^3+3x^2 \, dx\)
\(\textbf{8)}\) \(\displaystyle\int_{0}^{\pi} \, dx\)
\(\textbf{9)}\) \(\displaystyle\int_{1}^{2}5x^4 \, dx\)
\(\textbf{10)}\) \(\displaystyle\int_{0}^{1} 3x^2+4 \, dx\)
\(\textbf{11)}\) \(\displaystyle\int_{0}^{\pi/2} \sin(x) \, dx\)
\(\textbf{12)}\) \(\displaystyle\int_{-1}^{1} x^3 \, dx\)
\(\textbf{13)}\) \(\displaystyle\int_{0}^{2} (2x + 1) \, dx\)
\(\textbf{14)}\) \(\displaystyle\int_{-2}^{2} 4 \, dx\)
\(\textbf{15)}\) \(\displaystyle\int_{1}^{4} 3x \, dx\)
\(\textbf{16)}\) \(\displaystyle\int_{1}^{2} (6x^2 + 5x) \, dx\)
\(\textbf{17)}\) \(\displaystyle\int_{-1}^{1} x^5 \, dx\)
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\(\bullet\) Definite Integral Calculator (Symbolab)
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\(\bullet\text{ Trapezoidal Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{b-a}{2n}\left[f(a)+2f(x_1)+2f(x_2)+…+2fx_{n-1}+f(b)\right]…\)
\(\bullet\text{ Properties of Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}cf(x) \, dx=c\displaystyle \int_{a}^{b}f(x) \,dx…\)
\(\bullet\text{ Indefinite Integrals- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int x^n \, dx = \displaystyle \frac{x^{n+1}}{n+1}+C…\)
\(\bullet\text{ Indefinite Integrals- Trig Functions}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int \cos{x} \, dx=\sin{x}+C…\)
\(\bullet\text{ Definite Integrals}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{5}^{7} x^3 \, dx…\)
\(\bullet\text{ Integration by Substitution}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int (x^2+3)^3(2x) \,dx…\)
\(\bullet\text{ Area of Region Between Two Curves}\)
\(\,\,\,\,\,\,\,\,A=\displaystyle \int_{a}^{b}\left[f(x)-g(x)\right]\,dx…\)
\(\bullet\text{ Arc Length}\)
\(\,\,\,\,\,\,\,\,\displaystyle \int_{a}^{b}\sqrt{1+\left[f'(x)\right]^2} \,dx…\)
\(\bullet\text{ Average Function Value}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{1}{b-a} \int_{a}^{b}f(x) \,dx\)
\(\bullet\text{ Volume by Cross Sections}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Disk Method}\)
\(\,\,\,\,\,\,\,\,V=\displaystyle \int_{a}^{b}\left[f(x)\right]^2\,dx…\)
\(\bullet\text{ Cylindrical Shells}\)
\(\,\,\,\,\,\,\,\,V=2 \pi \displaystyle \int_{a}^{b} y f(y) \, dy…\)
In Summary
A definite integral is the limit of a sum, known as a Riemann sum. This sum represents the approximated area under a curve by dividing it into small rectangles and summing their areas. The definite integral is then calculated by taking the limit of the sum as the width of the rectangles approaches zero.
Definite integrals are an important mathematical tool that has numerous applications in fields such as physics, engineering, and economics. By learning about definite integrals, we are able to solve real-world problems involving the accumulation of quantities over a period of time or space.
Definite integrals are typically introduced in a Calculus course. In addition to their use in calculus, definite integrals have applications in other branches of mathematics, such as probability theory and mathematical physics. By learning about definite integrals, we open the door to exploring a wide range of mathematical concepts and applications.
Topics that use Definite Integrals
Displacement of a particle: The definite integral can be used to calculate the displacement of a particle moving in one dimension, given its velocity as a function of time. For example, if a car is moving at a velocity of \(v(x)\) per hour for 3 hours, its displacement can be calculated by finding the definite integral of the velocity function over that time interval.
Volume of a Solid of Revolution: The definite integral can be used to calculate the volume of a solid of revolution, which is formed by rotating a two-dimensional region around a line. For example, a cylinder can be created by rotating a rectangle around one of its sides. And the volume of this cylinder can be found using a definite integral.
Area under a curve: The definite integral can be used to calculate the area under a curve, which represents the area bounded by the curve, the x-axis, and two vertical lines. This can also be used for finding the area between two curves. This is often used in physics and engineering to calculate areas of irregular shapes, or to model real-world phenomena such as the flow of fluids or the distribution of temperatures.
Average value of a function: The definite integral can be used to calculate the average value of a function over a given interval. This can be useful when it is difficult to evaluate the function at all points in the interval.
Work done by a force: The definite integral can be used to calculate the work done by a force over a given distance. This is a key concept in mechanics and is used to understand the behavior of objects under the influence of forces.
Probability density functions: In probability and statistics, the definite integral of a probability density function over a given interval represents the probability that a random variable will take on a value within that interval. This is used to understand the likelihood of certain events occurring and to make predictions about future events.