Find the volume of the solid formed by rotating the region enclosed by
y=e^4x+3 y=0 x=0 x=0.7
this is my work
V = ∫ e^4x dx {between x=0 and x=0.7}
= 1/4 e^4x {between x=0 and x=0.7}
= 1/4 (e^2.8 - e^0)
= 1/4 (e^2.8 - 1)
however i am etting the wrong answer pls help!!!
What you have found is the area of the region.
But you wanted the volume.
We also have to assume you are rotating about the x-axis.
in general V = pi∫ y^2 dx
so you want
V = pi∫e^8x dx from 0 to .7
= (1/8)pi[e^8x] from 0 to .7
= (1.8)(pi(e^5.6 - e^0)
= 105.8
You better check my arithmetic.
Hi i checked the arithmitic and got 154.3699603 but this does not seem to be the right answer either what should I do??
Arggghhh!! I forgot the + 3 in the equation
try this
V = pi∫(e4x+x3)^2 dx from 0 to .7
= pi∫(e^8x + 6e^4x + 9)dx
= pi[(e^8x)/8 + (3/2)e^4x + 9x) from 0 to .7
= .... I will leave the arithmetic up to you, let me if it worked out this time.
To find the volume of the solid formed by rotating the region enclosed by the curves, you need to use the formula:
V = π∫[a,b] f(x)^2 dx
where [a,b] is the interval over which you are integrating and f(x) is the function describing the curve.
In this case, the given curves are y = e^4x + 3 and y = 0. To find the interval [a,b], we can solve for x when y = 0:
0 = e^4x + 3
e^4x = -3
Since the exponential function is always positive, there are no real solutions to this equation. Therefore, the interval of integration is from x = 0 to x = 0.7, as you correctly stated.
Now, let's calculate the volume using the correct integral setup:
V = π∫[0,0.7] (e^4x + 3)^2 dx
= π∫[0,0.7] (e^8x + 6e^4x + 9) dx
We can integrate this expression term by term:
V = π [1/8e^8x + 6/4e^4x + 9x] [from 0 to 0.7]
= π [1/8e^8(0.7) + 6/4e^4(0.7) + 9(0.7)] - π [1/8e^8(0) + 6/4e^4(0) + 9(0)]
= π [1/8e^5.6 + 6/4e^2.8 + 6.3] - π [0 + 0 + 0]
Simplifying further:
V = π [1/8e^5.6 + 3/2e^2.8 + 6.3]
Now you can calculate the volume using the appropriate value for π.