Answer to Question #205275 in Calculus for shane

Question #205275

1. The curve has an equation y = ex. Compute the area bounded by the curve from x = 0 to x = 1.

2. The loop of the curve has an equation of y2 = x(1 – x)2. Find the area enclosed by the loop of the curve.

3. Given the area in the first quadrant bounded by y2 = x, the line x = 4 and the x-axis. What is the volume generated when this area is revolved about the y-axis?

4. The region in the first quadrant which is bounded by the curve y2 = 4x, and the lines x = 4 and y = 0, is revolved about the x-axis. Locate the centroid of the resulting solid of revolution.

5. The region in the first quadrant, which is bounded by the curve x2 = 4y, the line x = 4, is revolved about the line x = 4. Locate the centroid of the resulting solid of revolution.




1
Expert's answer
2021-06-16T05:34:50-0400

Solution.

1.


S=01exdx=ex01=e1.S=\int_0^1 e^x dx=e^x|_0^1=e-1.

2.


S=201x1xdxS=2\int_0^1x\sqrt{1-x}dx

Find x1xdx,\int x\sqrt{1-x}dx, for this we do replacement

1x=u.1-x=u. Then x=1u,dx=du,x1x=(1u)u=uu32.x=1-u, -dx=du,\newline x\sqrt{1-x}=(1-u)\sqrt{u}=\sqrt{u}-u^{\frac{3}{2}}.

So, x1xdx=(uu32)du=25u5223u32=25(1x)5223(1x)32.\int x\sqrt{1-x}dx=-\int (\sqrt{u}-u^{\frac{3}{2}})du=\newline \frac{2}{5}u^{\frac{5}{2}}-\frac{2}{3}u^{\frac{3}{2}}=\newline \frac{2}{5}(1-x)^{\frac{5}{2}}-\frac{2}{3}(1-x)^{\frac{3}{2}}.

Therefore,

S=2(25(1x)5223(1x)32)01=815.S=2(\frac{2}{5}(1-x)^{\frac{5}{2}}-\frac{2}{3}(1-x)^{\frac{3}{2}})|_0^1=\newline \frac{8}{15}.

3.


Vy=2π04xxdx2π04x32dx2π25x5204=1285π=2535π.V_y=2π\int_0^4 x\sqrt{x}dx\newline 2π\int_0^4 x^{\frac{3}{2}}dx\newline 2π\frac{2}{5}x^{\frac{5}{2}}|_0^4=\newline \frac{128}{5}π=25\frac{3}{5}π.

4.



The volume of the body V formed by rotation around the axis Ox of the figure axb,y1(x)yy2(x),a\leq x\leq b,y_1(x)\leq y \leq y_2(x), , where y1(x) and y2(x) are continuous non-negative functions, is equal to a certain interval from the difference of the square of the functions yi(x) by the variable x:


Vx=πab(y22(x)y12(x))dxV_x=π\int_a^b(y_2^2(x)-y_1^2(x))dx

We will have

Vx=π04(2x)2dx=4πx2204=32π.V_x=π\int_0^4 (2\sqrt{x})^2dx=4π \frac{x^2}{2}|_0^4=\newline 32π.

In this case, the centroid lies on the x-axis. The formula to get the centroid of the figure is:

Vx=xdV.Vx^*=\int xdV.

32πx=044πx2dx32πx^*=\int_0^4 4πx^2dx

32x=4x330432x^*=\frac{4x^3}{3}|_0^4

From here, centroid x=83.x^*=\frac{8}{3}.

5.



The volume of the body V formed by rotation around the line x=m of the figure ayb,0xx(y)a\leq y\leq b,0\leq x\leq x(y) , where x(y) is a unique continuous function equal to the definite integral calculated by the formula


V=πab(a2x2(y))dyV=π\int_a^b(a^2-x^2(y))dy

Vx=4=π04((2y)2+43)dy=π(2y2+16y)04=32π.V_{x=4}=π\int_0^4(-(2\sqrt{y})^2+4^3)dy=π(-2y^{2}+16y)|_0^4=32π.

In this case, the centroid lies on the y-axis. The formula to get the centroid of the figure isVy=ydV.Vy^*=\int ydV.

Vy=04yπ2ydy=1285π.Vy^*=\int_0^4yπ2\sqrt{y}dy=\frac{128}{5}π.

From here y=45.y^*=\frac{4}{5}.

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