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...
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
What I did was:
INT (-1,0) (4x+4)(5+x) dx
INT 4x^+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
Well, calculating the total mass of a triangular region can be a bit tricky, but fear not, I'm here to help you out with a touch of humor!
To calculate the total mass of the triangular region, we can break it down into simpler shapes. In this case, let's break it down into two right triangles.
The first right triangle is defined by the coordinates (-1,0), (0,4), and (-1,4). The base of this triangle is 1 cm, and the height is 4 cm.
The second right triangle is defined by the coordinates (0,4), (1,0), and (1,4). The base of this triangle is also 1 cm, and the height is 4 cm.
Now, we can calculate the mass of each of these triangles separately. Let's start with the first one.
For the first right triangle, the area is (1/2) * base * height = (1/2) * 1 cm * 4 cm = 2 cm^2.
The density function for this triangle is (x) = 5 + x grams/cm^2. So, plugging in the area, we get: mass = (5 + 2) grams = 7 grams.
For the second right triangle, the area is also 2 cm^2, and the density function is still (x) = 5 + x grams/cm^2. Again, plugging in the area, we get: mass = (5 + 2) grams = 7 grams.
Finally, we add the masses of both triangles together to get the total mass: 7 grams + 7 grams = 14 grams.
So, the total mass of the triangular region is 14 grams.
Hope this explanation has brought a smile to your face while solving this problem!
To find the total mass of the triangular region, you need to calculate the area of the triangle first, and then integrate the density function over that region.
To calculate the area of the triangle, you can use the formula for the area of a triangle given its vertices:
Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
In this case, the vertices of the triangle are (-1, 0), (0, 4), and (1, 0). Plugging these coordinates into the formula, we get:
Area = 1/2 * |(-1)(4 - 0) + (0)(0 - 0) + (1)(0 - 4)|
Area = 1/2 * |-4 + 0 + 0|
Area = 1/2 * |-4|
Area = 2 cm^2
Now that we have the area of the triangle, we can integrate the density function over this region to find the total mass. The density function is given as (x) = 5 + x grams/cm^2.
The integral for mass is given by:
Total Mass = ∫(Density) dA
where dA represents the infinitesimal area element.
In this case, since the density is a function of position (x), we can express the total mass as:
Total Mass = ∫(5 + x) dA
To integrate over the triangular region, you can use the coordinates (-1, 0), (0, 4), and (1, 0) to set up the appropriate limits of integration.
Total Mass = ∫∫(5 + x) dxdy
where the limits of integration for x and y are as follows:
x: -1 to 1
y: 0 to (4 - 4x)
Evaluating this double integral will give you the total mass of the triangular region.