5.2 The Definite Integral Flashcards
In this section we will explore two things. We will consider what happens as we make the widths of the sub intervals smaller, and we will consider what happens when the function can be positive, zero, and/or negative on the desired interval. (19 cards)
What is the definition of the definite integral of f(x) over the interval [a, b]?
∫_a^b f(x) dx = lim (n→∞) Σ f(x*) Δx_i
This represents the limit of Riemann sums as the number of subintervals increases without bound.
What is the difference between an antiderivative and a definite integral?
An antiderivative is a family of functions; a definite integral is a number.
Antiderivative: ∫ f(x) dx = F(x) + C; Definite integral: ∫ f(x) dx = I.
What is the integral sign used for in calculus?
The integral sign (∫) is used to denote integration.
It represents the operation of finding the integral of a function.
What is the variable of integration in the integral ∫ f(x) dx?
x
This denotes the variable with respect to which the function f(x) is integrated.
What are the upper and lower limits of integration in the integral ∫_a^b f(x) dx?
Upper limit: b; Lower limit: a
These limits define the interval over which the function is integrated.
What is a Riemann sum?
A Riemann sum is an approximation of the area under a curve using a finite number of rectangles.
It is calculated by summing the areas of rectangles that approximate the area under the curve.
Fill in the blank: The process of making the widths of the subintervals smaller in a Riemann sum leads to the ______.
definite integral
True or False: A definite integral can be negative.
True
This occurs when the area under the curve is below the x-axis.
What is meant by ‘signed area’ in the context of definite integrals?
Signed area refers to the net area that can be positive, negative, or zero, depending on the position of the function relative to the x-axis.
What is the formula for the area of a trapezoid?
Area = 1/2 * (base1 + base2) * height
This formula can be applied in evaluating certain definite integrals.
What is the significance of geometry in computing areas between functions and the x-axis?
Geometry can provide exact values for areas represented by definite integrals.
It allows for alternative methods of calculating integrals beyond numerical or analytical approaches.
What happens when the integrand is below the x-axis?
The Riemann sum will yield negative contributions to the total area.
The areas of rectangles representing the integrand will be subtracted from the total area.
What does it mean to compute the net signed area?
It means calculating the area under the curve taking into account whether it lies above or below the x-axis.
What is the notation for the definite integral of a function f(x) from a to b?
∫_a^b f(x) dx
List the components of the definite integral notation ∫_a^b f(x) dx.
- Integral sign (∫)
- Integrand (f(x))
- Lower limit (a)
- Upper limit (b)
- Differential (dx)
How do you compute a definite integral using integral properties?
By applying known values of definite integrals and properties such as linearity and additive intervals.
Fill in the blank: The area between the graph of f(x) = -2x + 4 and the x-axis over the interval [0, 9] can be computed as a ______.
definite integral
What is the relationship between the number of subintervals and the accuracy of a Riemann sum?
As the number of subintervals increases, the accuracy of the Riemann sum improves.
