Integral
Definition
An integral is a mathematical concept used to calculate areas, volumes, and accumulated quantities by finding antiderivatives or sums of infinitesimal values.
Analogy
Imagine an integral as a tool that measures how much water flows through a pipe over time. By integrating the flow rate function over time, you can determine the total amount of water that has passed through.
Related terms
Derivative: The derivative of a function represents its rate of change at any given point.
Riemann sum: An approximation method for calculating definite integrals by dividing an interval into subintervals and summing up areas under curves within those subintervals.
Fundamental theorem of calculus: States that differentiation and integration are inverse operations, connecting derivatives with integrals.
"Integral" appears in:
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Additional resources (2)
Practice Questions (20+)
- Which type of Riemann Sum will always result in an overestimate of the integral for increasing functions?
- Which type of Riemann Sum will always result in an underestimate of the integral for decreasing functions?
- The estimate obtained from a trapezoidal Riemann Sum will be closer to the true value of the integral compared to which other types of Riemann Sums?
- Approximate the value of $\int_{1}^{6} f(x) , dx$ using 5 equal subintervals, where $f(x) = 2x - 1$. Calculate the trapezoidal Riemann Sum for this integral.
- Which type of integral is indicated by answer options that start off as polynomials and end with a natural log term?
- What should be done first when attempting to complete the square for an integral?
- What type of answer options indicate a completing the square integral?
- Evaluate the integral: ∫ 1/ (x^2 + 4x + 8) dx
- Generally speaking, should the portion of the integral that is a polynomial be your f(x) or your g(x)?
- Generally speaking, should the portion of the integral that contains a trigonometric function be your f(x) or your g(x)?
- The integral, $\int_{0}^{4} (3 + 4t) , dt$ represents the accumulation of the function $(3 + 4t)$ over the interval $[0, 4]$. Evaluate the integral.
- The integral, $\int_{x}^{x^2} \frac{\sin(t)}{t} , dt$ represents the accumulation of the function $\frac{\sin(t)}{t}$ over the interval $[0, x]$. What is the derivative $F'(x)$ of this function?
- What part of the function in the integral ∫sqrt(3x+7)dx should u be set equal to in order to use a u-substitution to solve the integral?
- Rewrite the integral ∫sqrt(3x+7)dx with an appropriate u-substitution.
- What is the value of the integral 1/3∫sqrt(u) du?
- What is an appropriate choice of u for the integral ∫(0 to 2) (x(sqrt(3x^2 + 7) dx?
- If u = 3x^2 + 7, and the original bounds of the integral are 0 and 2, what are the bounds in terms of u?
- The graphs of y = x^2 -4 and y = 2x - x^2 create a bounded area that is the base of a solid. This solid has cross sections that are perpendicular to the 𝑥-axis and form squares. What are the bounds of the integral for the volume of this solid?
- The base of a solid with square cross sections is bounded by y = \sqrt{x-1}, x = 3, y = 0. What are the numerical bounds for the integral that can be used to find the volume of this solid when cross sections are taken perpendicular to the x-axis?
- The base of a solid with square cross sections is bounded by y = \sqrt{x-1}, x = 3, y = 0. Which integral can be used to find the volume of this solid when cross-sections are taken perpendicular to the x-axis?
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