Revise later
SpaceTo flip
If confident
All Flashcards
What is the formula for the area between two curves y=f(x) and y=g(x) from x=a to x=b?
$\text{Area} = \int_{a}^{b} |f(x) - g(x)| , dx$
How do you calculate the area between curves when f(x) > g(x) on [a, c] and g(x) > f(x) on [c, b]?
$\text{Area} = \int_{a}^{c} (f(x) - g(x)) , dx + \int_{c}^{b} (g(x) - f(x)) , dx$
What is the general form of the antiderivative of $x^n$?
$\int x^n , dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$
What is the formula for the area between two curves if the integral is with respect to y?
$\text{Area} = \int_{c}^{d} |f(y) - g(y)| , dy$, where $x=f(y)$ and $x=g(y)$
What is the formula for the antiderivative of a polynomial?
$\int (ax^n + bx^m) , dx = a\frac{x^{n+1}}{n+1} + b\frac{x^{m+1}}{m+1} + C$
What is the formula for finding intersection points of two curves?
Set $f(x) = g(x)$ and solve for $x$.
How do you express the definite integral with limits of integration?
$\int_{a}^{b} f(x) , dx = F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$.
What is the formula to find the area between curves using vertical slices?
$\text{Area} = \int_{a}^{b} (f(x) - g(x)) , dx$, where $f(x)$ is the top function and $g(x)$ is the bottom function.
How do you represent the area between two curves when integrating with respect to x?
$\text{Area} = \int_{a}^{b} |f(x) - g(x)| , dx$
What is the formula for the area between curves when integrating with respect to y?
$\text{Area} = \int_{c}^{d} |h(y) - k(y)| , dy$, where $h(y)$ is the right function and $k(y)$ is the left function.
Explain the concept of finding the area between intersecting curves.
It involves integrating the absolute difference between two functions over an interval, accounting for changes in which function is 'on top'.
Why is absolute value important when finding the area between curves?
It ensures that the area is always positive, regardless of whether f(x) > g(x) or g(x) > f(x).
Describe the role of intersection points in finding the area between curves.
Intersection points define the intervals over which different functions act as the 'top' or 'bottom' curve, requiring separate integrals.
Explain the relationship between definite integrals and area.
The definite integral calculates the signed area between a curve and the x-axis. Area between curves uses this to find the area between two functions.
What does it mean for a region to be 'negative' when finding the area between curves?
It means that the function being integrated is below the x-axis, contributing a negative value to the integral unless absolute value is used.
Why is graphing the functions helpful when finding the area between curves?
Graphing helps visualize which function is on top and identify the intersection points, aiding in setting up the correct integrals.
Explain the concept of 'vertical slices' in area calculation.
Approximating the area by dividing it into thin vertical rectangles, where the height is the difference between the functions and the width is dx.
What is the significance of setting two equations equal?
It's the method to find the x-coordinates where the two curves intersect, which are essential for defining the integration intervals.
Explain why the order of subtraction matters when setting up the integral.
The order determines whether the result is positive or negative; using absolute value corrects for this by ensuring a positive area.
Describe the role of the antiderivative in evaluating the definite integral.
The antiderivative is used to evaluate the definite integral at the bounds of integration, giving the net change over the interval.
How do you find the area between y = x^2 and y = x^3 from x = 0 to x = 1?
1) Determine intersection points (0, 1). 2) Identify top/bottom functions (x^2 > x^3). 3) Integrate: $\int_{0}^{1} (x^2 - x^3) , dx$. 4) Evaluate: [x^3/3 - x^4/4]_0^1 = 1/12.
Steps to find the area between curves intersecting at more than two points.
1) Find intersection points. 2) Graph functions. 3) Choose integration approach (dx or dy). 4) Set up integral(s) with correct intervals. 5) Evaluate integral(s).
How to determine the limits of integration when finding the area between curves?
Find the x-coordinates (or y-coordinates if integrating with respect to y) of the intersection points of the curves.
How do you handle multiple intersection points when finding the area between curves?
Split the integral into multiple integrals, each over an interval defined by consecutive intersection points, ensuring the correct order of subtraction within each interval.
How do you decide whether to integrate with respect to x or y?
Choose the variable that simplifies the integral setup; if the functions are easily expressed as functions of y, integrate with respect to y; otherwise, integrate with respect to x.
How to find the area between $f(x) = x^2$ and $g(x) = 2x$ from $x = 0$ to $x = 3$?
1) Find intersection points: $x^2 = 2x Rightarrow x = 0, 2$. 2) Split integral: $\int_{0}^{2} (2x - x^2) , dx + \int_{2}^{3} (x^2 - 2x) , dx$. 3) Evaluate each integral.
How do you find the area between $y = x$ and $y = x^3$?
1) Find intersection points: $x = x^3 Rightarrow x = -1, 0, 1$. 2) Set up integrals: $\int_{-1}^{0} (x^3 - x) , dx + \int_{0}^{1} (x - x^3) , dx$. 3) Evaluate the integrals.
How do you approach a problem asking for the area between curves given only the equations?
1) Find intersection points. 2) Sketch a graph (optional but recommended). 3) Determine which function is on top. 4) Set up and evaluate the definite integral.
How do you find the area between $y = x^2 - 4$ and $y = 4 - x^2$?
1) Intersection points: $x^2 - 4 = 4 - x^2 Rightarrow x = -2, 2$. 2) Set up integral: $\int_{-2}^{2} (4 - x^2 - (x^2 - 4)) , dx$. 3) Evaluate the integral.
How do you find the area when the curves are given as functions of y?
1) Express curves as x = f(y) and x = g(y). 2) Find intersection points in terms of y. 3) Integrate with respect to y: $\int_{c}^{d} |f(y) - g(y)| , dy$.