I know I posted this problem earlier but I don't think it will be seen anymore since it's almost at the end of the page:
Find the total mass of the triangular region with coordinates (-1,0),(0,4),and (1,0). All lengths are in centimeters, and the density of the region is (x)=5+x grams/c^m2.
I know that to get the total mass i have to do density* lenght and ingegrate, but I don't know how to to it for a triangle...
Calculus - bobpursley, Wednesday, October 26, 2011 at 9:56pm
draw the triangle.
Notice the base is horizontal frm -1,0 to 1,0
write the equation for the two legs
left leg: y=mx+b where m= (4/1)=4
y=4x+b
4=4*0+b or b=4
integrating the left side..
mass=INT y*(5+x)dx fro x=-1 to 0 do that inegral. Now on the right side, the leg equation is y=-4x+4
same equation as above, integrate from x=0 to 1
be certain in each area to use as y either (-4x+4) or y=(4x+4)
add the two masses
Calculus - Caroline, Wednesday, October 26, 2011 at 10:38pm
What I did was:
INT (-1,0) (4x+4)(5+x) dx
INT 4x^2+36x+20
I integrated and got (4/3)x^3+18x^2+20x
and evaluated from -1 to 0 and got 10/3 g and then from 0 to 1 and got 118/3 grams added and got 128/3 g and that's not the answer. What am I doing wrong?
(4x+4)(5+x) = 4x^2 + 24x + 20
I get the same answer...
I meant the same final answer, because now I get 28/3 and 100/3
Hmm. I got 28/3 for the left, 32/3 on the right = 20 total
(-4x+4)(x+5) = -4x^2 - 16x + 20
Integral is -4/3 x^3 - 8x^2 + 20x
at x=1 that's -4/3 - 8 + 20 = 32/3
To find the total mass of the triangular region, you need to calculate the integral of the density function over the triangular region.
First, let's analyze the triangle. The base of the triangle is the line segment connecting the points (-1, 0) and (1, 0), which is a horizontal line. The length of the base is 2 centimeters.
The two legs of the triangle are lines with different slopes but are both linear. The left leg of the triangle has a slope of 4, and its equation is y = 4x + 4. The right leg of the triangle has a slope of -4, and its equation is y = -4x + 4.
To find the mass of the region, we need to split it into two parts - the left side and the right side of the triangle. We will calculate the mass for each side separately and then add them together.
For the left side of the triangle:
1. Calculate the integral of the density function (5 + x) multiplied by the height of the left side, which is given by the equation y = 4x + 4.
∫[from -1 to 0] (4x + 4)(5 + x) dx
2. Integrate the function with respect to x:
∫[from -1 to 0] (20x + 4x^2 + 20 + 4x) dx
3. Evaluate the integral:
= [(10x^2 + 2x^3 + 20x + 2x^2)] [from -1 to 0]
= [10(0)^2 + 2(0)^3 + 20(0) + 2(0)^2] - [10(-1)^2 + 2(-1)^3 + 20(-1) + 2(-1)^2]
= [0 + 0 + 0 + 0] - [10 + (-2) - 20 + 2]
= -8 grams
For the right side of the triangle:
1. Calculate the integral of the density function (5 + x) multiplied by the height of the right side, which is given by the equation y = -4x + 4.
∫[from 0 to 1] (-4x + 4)(5 + x) dx
2. Integrate the function with respect to x:
∫[from 0 to 1] (-20x + 4x^2 + 20 - 4x) dx
3. Evaluate the integral:
= [(-10x^2 + 2x^3 + 20x - 2x^2)] [from 0 to 1]
= [(-10(1)^2 + 2(1)^3 + 20(1) - 2(1)^2] - [(-10(0)^2 + 2(0)^3 + 20(0) - 2(0)^2]
= [-10 + 2 + 20 - 2] - [0 + 0 + 0 - 0]
= 10 grams
Now, add the masses from the left and right sides to find the total mass of the triangular region:
(-8 grams) + (10 grams) = 2 grams
Therefore, the total mass of the triangular region is 2 grams.