set up sums of integrals that can be used to find the area of the region bounded by the graphs of the equations by integrating with respect to y
y=square root of x; y= -x, x=1, x=4
∫[-4,-1] 4-(-y) dy
+∫[-1,1] 4-(1) dy
+∫[1,2] 4-(y^2) dy
The two verticals x=1 and x=4 intersect
y =√x at A(1,1) and B(4,2) and they intersect
y = -x at D(1,-1) and C(4,-4)
from A draw a horizontal to hit BC at (4,1)
So we have a trapezoid, for which we don't need Calculus to find its area
AD = 2, EC = 5 , and the height between them is 3
Area of trap AECD = (1/2)(2+5)(3) = 21/2 square units
the area of the region outlined by ABE is
∫(4 - y^2) dy from 1 to 2
= [4y - (1/3)y^3] from 1 to 2
= (8 - 8/3) - (4 - 1/3)
= 5/3
total area = 21/2 + 5/3 = 73/6
I don't understand why they would ask that it be done by integrating with respect to y, when integrating it with respect to x would have been so much easier.
Area = ∫(x^(1/2) + x) dx from 1 to 4
= [(2/3)x^(3/2) + (1/2)x^2] from 1 to 4
= (16/3 + 8) - (2/3 + 1/2)
= 73/6
(that took 4 lines)
To find the area of the region bounded by the graphs of the equations y = √x, y = -x, and the vertical lines x = 1 and x = 4 by integrating with respect to y, we'll need to split the region into two parts: the portion above the x-axis (where y is positive) and the portion below the x-axis (where y is negative).
Let's start with the portion above the x-axis.
1. Determine the limits of integration for y.
The upper limit of integration for y is given by the equation y = √x.
The lower limit of integration for y is given by the equation y = -x.
Both equations intersect at (0, 0), so the limits of integration for y are 0 to √4, which simplifies to 0 to 2.
2. Set up the integral to find the area.
Since we are integrating with respect to y, we need to express x in terms of y.
From the equation y = √x, we can solve for x as x = y^2.
The integral for the portion above the x-axis is:
∫[0 to 2] (upper function - lower function) dx
∫[0 to 2] (√x - (-x)) dx
∫[0 to 2] (x + x) dx
∫[0 to 2] 2x dx
Next, let's consider the portion below the x-axis.
1. Determine the limits of integration for y.
The upper limit of integration for y is given by the equation y = -x.
The lower limit of integration for y is y = -√4
2. Set up the integral to find the area.
Similar to the previous step, we need to express x in terms of y.
From the equation y = -x, we can solve for x as x = -y.
The integral for the portion below the x-axis is:
∫[-2 to 0] (upper function - lower function) dx
∫[-2 to 0] (-y - (-y^2)) dx
∫[-2 to 0] (y^2 - y) dx
To find the total area of the region bounded by the two graphs, we add the areas of both portions together:
Total area = ∫[0 to 2] 2x dx + ∫[-2 to 0] (y^2 - y) dx
Evaluating these integrals will give you the final numerical result.