Answer:
see solution
Step-by-step explanation:
equation: y = 1 - 3x
when x is -2:
y = 1 -3 * -2 = y = 1+6 = y = 7
coordinates = (x,y) = (-2,7)
when x is -1:
y = 1-3*-1 = y = 1+3 = y = 4
coordinates = (x,y) = (-1,4)
when x is 0:
y = 1 - 3 * 0 = y = 1-0 = y =1
coordinates = (x,y) = (0,1)
when x is 1
y = 1-3*1 = y = 1-3 = y = -2
coordinates = (x,y) = (1,-2)
When x is 2
y = 1-3*2 = y = 1-6 = y = -5
coordinates = (x,y) = (2,-5)
when x is 3
y = 1-3*3 = y = 1-9 = y = -8
coordinates = (x,y) = (3,-8)
Plot these points on the graph and join up the dots to see the line.
bye
Answer:
Step-by-step explanation:
![\\ \int\limits^{a}_{0} \int\limits^{x}_{0} \int\limits^{x+y}_{0} {e^{x+y+z}} \, dzdydx \\\\=\int\limits^{a}_{0} \int\limits^{x}_{0} [\int\limits^{x+y}_{0} {e^{x+y}e^z} \, dz]dydx \\\\\\=\int\limits^{a}_{0} \int\limits^{x}_{0} [e^{x+y}\int\limits^{x+y}_{0} {e^z} \, dz]dydx\\\\=\int\limits^{a}_{0} \int\limits^{x}_{0} [e^{x+y}e^z\Big|_0^{x+y}]dydx \\\\\\=\int\limits^{a}_{0} \int\limits^{x}_{0} [e^{x+y}e^{x+y}-e^{x+y}]dydx \\\\\\=\int\limits^{a}_{0} \int\limits^{x}_{0} e^{2x+2y}-e^{x+y}dydx \\\\\\](https://tex.z-dn.net/?f=%5C%5C%20%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%20%5Cint%5Climits%5E%7Bx%7D_%7B0%7D%20%5Cint%5Climits%5E%7Bx%2By%7D_%7B0%7D%20%7Be%5E%7Bx%2By%2Bz%7D%7D%20%5C%2C%20dzdydx%20%5C%5C%5C%5C%3D%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%20%5Cint%5Climits%5E%7Bx%7D_%7B0%7D%20%5B%5Cint%5Climits%5E%7Bx%2By%7D_%7B0%7D%20%7Be%5E%7Bx%2By%7De%5Ez%7D%20%5C%2C%20dz%5Ddydx%20%5C%5C%5C%5C%5C%5C%3D%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%20%5Cint%5Climits%5E%7Bx%7D_%7B0%7D%20%5Be%5E%7Bx%2By%7D%5Cint%5Climits%5E%7Bx%2By%7D_%7B0%7D%20%7Be%5Ez%7D%20%5C%2C%20dz%5Ddydx%5C%5C%5C%5C%3D%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%20%5Cint%5Climits%5E%7Bx%7D_%7B0%7D%20%5Be%5E%7Bx%2By%7De%5Ez%5CBig%7C_0%5E%7Bx%2By%7D%5Ddydx%20%5C%5C%5C%5C%5C%5C%3D%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%20%5Cint%5Climits%5E%7Bx%7D_%7B0%7D%20%5Be%5E%7Bx%2By%7De%5E%7Bx%2By%7D-e%5E%7Bx%2By%7D%5Ddydx%20%5C%5C%5C%5C%5C%5C%3D%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%20%5Cint%5Climits%5E%7Bx%7D_%7B0%7D%20e%5E%7B2x%2B2y%7D-e%5E%7Bx%2By%7Ddydx%20%5C%5C%5C%5C%5C%5C)
![\\=\int\limits^{a}_{0} [\int\limits^{x}_{0} e^{2x}e^{2y}-e^{x+y}dy]dx \\\\\\=\int\limits^{a}_{0} [\int\limits^{x}_{0} e^{2x}e^{2y}dy- \int\limits^{x}_{0}e^{x}e^{y}dy]dx \\\\\\u=2y\\du=2dy\\dy=\frac{1}{2}du\\\\\\=\int\limits^{a}_{0} [\frac{e^{2x}}{2}\int e^{u}du- e^x\int\limits^{x}_{0}e^{y}dy]dx \\\\\\=\int\limits^{a}_{0} [\frac{e^{2x}}{2}\cdot e^{2y}\Big|_0^x- e^xe^{y}\Big|_0^x]dx \\\\\\=\int\limits^{a}_{0} [\frac{e^{2x+2y}}{2} - e^{x+y}\Big|_0^x]dx \\\\](https://tex.z-dn.net/?f=%5C%5C%3D%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%20%5B%5Cint%5Climits%5E%7Bx%7D_%7B0%7D%20e%5E%7B2x%7De%5E%7B2y%7D-e%5E%7Bx%2By%7Ddy%5Ddx%20%5C%5C%5C%5C%5C%5C%3D%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%20%5B%5Cint%5Climits%5E%7Bx%7D_%7B0%7D%20e%5E%7B2x%7De%5E%7B2y%7Ddy-%20%5Cint%5Climits%5E%7Bx%7D_%7B0%7De%5E%7Bx%7De%5E%7By%7Ddy%5Ddx%20%5C%5C%5C%5C%5C%5Cu%3D2y%5C%5Cdu%3D2dy%5C%5Cdy%3D%5Cfrac%7B1%7D%7B2%7Ddu%5C%5C%5C%5C%5C%5C%3D%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%20%5B%5Cfrac%7Be%5E%7B2x%7D%7D%7B2%7D%5Cint%20e%5E%7Bu%7Ddu-%20e%5Ex%5Cint%5Climits%5E%7Bx%7D_%7B0%7De%5E%7By%7Ddy%5Ddx%20%5C%5C%5C%5C%5C%5C%3D%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%20%5B%5Cfrac%7Be%5E%7B2x%7D%7D%7B2%7D%5Ccdot%20e%5E%7B2y%7D%5CBig%7C_0%5Ex-%20e%5Exe%5E%7By%7D%5CBig%7C_0%5Ex%5Ddx%20%5C%5C%5C%5C%5C%5C%3D%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%20%5B%5Cfrac%7Be%5E%7B2x%2B2y%7D%7D%7B2%7D%20-%20e%5E%7Bx%2By%7D%5CBig%7C_0%5Ex%5Ddx%20%5C%5C%5C%5C)
![\\=\int\limits^{a}_{0} [\frac{e^{4x}}{2} - e^{2x}-\frac{e^{2x}}{2} + e^{x}]dx \\\\\\=\int\limits^{a}_{0} \frac{e^{4x}}{2} -\frac{3e^{2x}}{2} + e^{x}dx \\\\\\=\int\limits^{a}_{0} \frac{e^{4x}}{2}dx -\int\limits^{a}_{0}\frac{3e^{2x}}{2}dx + \int\limits^{a}_{0}e^{x}dx \\\\\\u_1=4x\\du_1=4dx\\dx=\frac{1}{4}du_1\\\\\u_2=2x\\du_2=2dx\\dx=\frac{1}{2}du_2\\\\\\=\frac{1}{8}\int e^{u_1}du_1 -\frac{3}{4}\int e^{u_2}du_2 + \int\limits^{a}_{0}e^{x}dx \\\\\\](https://tex.z-dn.net/?f=%5C%5C%3D%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%20%5B%5Cfrac%7Be%5E%7B4x%7D%7D%7B2%7D%20-%20e%5E%7B2x%7D-%5Cfrac%7Be%5E%7B2x%7D%7D%7B2%7D%20%2B%20e%5E%7Bx%7D%5Ddx%20%5C%5C%5C%5C%5C%5C%3D%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%20%5Cfrac%7Be%5E%7B4x%7D%7D%7B2%7D%20-%5Cfrac%7B3e%5E%7B2x%7D%7D%7B2%7D%20%2B%20e%5E%7Bx%7Ddx%20%5C%5C%5C%5C%5C%5C%3D%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%20%5Cfrac%7Be%5E%7B4x%7D%7D%7B2%7Ddx%20-%5Cint%5Climits%5E%7Ba%7D_%7B0%7D%5Cfrac%7B3e%5E%7B2x%7D%7D%7B2%7Ddx%20%2B%20%5Cint%5Climits%5E%7Ba%7D_%7B0%7De%5E%7Bx%7Ddx%20%5C%5C%5C%5C%5C%5Cu_1%3D4x%5C%5Cdu_1%3D4dx%5C%5Cdx%3D%5Cfrac%7B1%7D%7B4%7Ddu_1%5C%5C%5C%5C%5Cu_2%3D2x%5C%5Cdu_2%3D2dx%5C%5Cdx%3D%5Cfrac%7B1%7D%7B2%7Ddu_2%5C%5C%5C%5C%5C%5C%3D%5Cfrac%7B1%7D%7B8%7D%5Cint%20e%5E%7Bu_1%7Ddu_1%20-%5Cfrac%7B3%7D%7B4%7D%5Cint%20e%5E%7Bu_2%7Ddu_2%20%2B%20%5Cint%5Climits%5E%7Ba%7D_%7B0%7De%5E%7Bx%7Ddx%20%5C%5C%5C%5C%5C%5C)
Sorry if that took a while to finish. I am in AP Calculus BC and that was my first time evaluating a triple integral. You will see some integrals and evaluation signs with blank upper and lower boundaries. I just had my equation in terms of u and didn't want to get any variables confused. Hope this helps you. If you have any questions let me know. Have a nice night.
The and answer is going to be 3 because when you multiply 5 and 3 you get 15
Im thinking both shorter sides are 5cm, and the 2 longer sides are 8cm each... add them all together and you get 26cm
