cmab said:
OlderDan said:
p53ud0 dr34m5 said:
cmab said:
The formula for finding the volume of a solid bounded by a curve y=f(x) and two lines x=a and x=b is given by V = ∫abπ[f(x)]2 dx. This is known as the disk or washer method.
First, we need to identify the limits of integration. Since the curve y=sinx is bounded by the y axis (x=0) and the line y=1 (x=π), the limits of integration are 0 and π. The integral will be V = ∫0ππ[sin(x)]2 dx.
Yes, you can also use the shell method to find the volume of this solid. The formula for the shell method is V = ∫ab2πx[f(x)-g(x)] dx, where f(x) is the upper curve and g(x) is the lower curve. In this case, f(x)=1 and g(x)=sin(x).
To solve the integral V = ∫0ππ[sin(x)]2 dx, you can use integration by parts or trigonometric identities. The final answer should be V = (2π-4)/3 ≈ 0.81 units3.
Yes, the disk or washer method and the shell method can be used to find the volume of any solid bounded by a curve and two lines, as long as the curve is continuous and the lines do not intersect the curve. The limits of integration and the formula may vary depending on the specific shape of the solid.