Definite Integrals
Instructions
- First, you should watch the concept videos below explaining the topics in the section.
- Next, you should attempt to solve the exercises. You can check your answers and watch videos explaining the exercises.
- When you have finished the material below, you can start on the next section or return to the main workshop page.
Concepts
- Properties of definite integrals
- Integral comparison properties
- The Fundamental Theorem of Calculus
- Evaluating definite integrals
- Finding the area between curves
Exercises
Directions: You should attempt to solve the problems first and then watch the video to see the solution.
1. If \(\displaystyle \int_{-1}^5 f(x)\, dx=-3\) and \(\displaystyle \int_{2}^5 f(x)\, dx =4,\) find \(\displaystyle \int_{-1}^2 f(x)\,dx.\)
2. Compare \(\displaystyle \int_{-1}^3\left(10-x^2\right) \, dx\) and \(\displaystyle \int_{-1}^3 (-x) \, dx.\) Determine the smallest and largest possible values for each of these definite integrals.
3. Evaluate the following definite integral exactly.
\[ \displaystyle \int_2^5 \frac{6x^3-x^2+\sqrt{x}}{x^3}\,dx\]
\[ \displaystyle \int_2^5 \frac{6x^3-x^2+\sqrt{x}}{x^3}\,dx\]
4. Evaluate the following definite integral exactly.
\[ \int_{-1}^1 \left(4x^2-8x-3\right)x^7\, dx\]
\[ \int_{-1}^1 \left(4x^2-8x-3\right)x^7\, dx\]
5. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of \( \displaystyle F(x)=\int_8^x \left( \sqrt{t}+5t^2\right) \, dt.\) Verify your answer using Part 2 of the Fundamental Theorem of Calculus.
6. Find the area of the region between the curves \(y=f(x)=5\) and \(y=g(x)=2x^2+5\) on the interval \([0,3].\)
7. Find the area of the region bounded by the curve \(y=f(x)=x^3+x^2-6x\) and the \(x\)-axis.