Figure is attached below:
Answer:
[tex]100.04\pi[/tex]
Step-by-step explanation:
Given Data:
triangle vertices= A,B,C=(0,0), (7,0), (0,7)
Required: Find Volume of the solid=V=?
For this type of question we need to use the following formula of slicing method
[tex]\int\limits^a_bA {(x)} \, dx................Eq(1)[/tex]
where a and b are limits which shows that solid extends from [tex]x=a\ to\ x=b[/tex] with the known cross section area [tex]A(x)[/tex] perpendicular to x-axis.
From the Figure.2 we can see that limits are from [tex]x=0[/tex] to [tex]x=7[/tex] and the red line is the diameter of the given semi circle.
We can see that if x increases then the diameter decreases along the hypotenuse thus diameter of the semi circle[tex]=d=[/tex] [tex]7-x[/tex]
Radius is half of the diameter, so Radius of semi circle[tex]=r=\frac{7-x}{2}[/tex]
Here, Area [tex]A(x)[/tex] is the area of the semi circle
so [tex]A(x)=\frac{1}{2}(\pi ) r^{2}[/tex]
Putting values, we get
[tex]A(x)=\frac{1}{2}(\pi ) (\frac{7-x}{2}) ^{2}[/tex]
Putting values in Eq(1), we get
[tex]V=\int\limits^7_0 {\frac{1}{2}(\pi ) (\frac{7-x}{2}) ^{2}} \, dx[/tex]
[tex]V=\frac{\pi }{2}\int\limits^0_7 {(\frac{7-x}{2} )^{2} } \, dx[/tex]
[tex]V=\frac{\pi }{2}\int\limits^0_7 {(\frac{49+x^{2} -2(7)(x)}{4} ) } \, dx[/tex] expanding formula ([tex](a+b)^{2}=a^{2}+b^{2}+2ab[/tex])
[tex]V=\frac{\pi }{8}\int\limits^7_0 ({49+x^{2} -14x)} \, dx[/tex]
[tex]V=\frac{\pi }{8}(\int\limits^7_0 {49} \, dx+\int\limits^7_0 {x^{2} } \, dx +\int\limits^7_0 {14x} \, dx )[/tex]
[tex]V=\frac{\pi }{8}(49x+\frac{x^3}{3}+\frac{14x^2}{2}) (for\ x=0\ to\ 7)[/tex]
Putting [tex]x=0[/tex] we get everything zero that's why only have expression for value of [tex]x=7[/tex] as:
[tex]V=\frac{\pi }{8}(49*7+\frac{7^{3}}{3}+14*\frac{7^2}{2} )[/tex]
[tex]V=\frac{\pi }{8}(343+114.33+343)[/tex]
[tex]V=\frac{\pi }{8}(800.33)\\\\ V=100.04\pi }[/tex]
