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Consider the curve f(x)=x^4 between x = -1 and x = 4.
a)What is the volume obtained by revolving the area under the curve around the x-axis?
b)What is the volume obtained by revolving the area under the curve (i.e. between f(x) and the x-axis) around the line y = -5?
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines.
y=x^2 ,y=4x-x^2
the x- axis b) the line y=6
a) To find the volume obtained by revolving the area under the curve f(x)=x^4 around the x-axis, you can use the method of cylindrical shells. Here are the steps:
Step 1: Determine the limits of integration. In this case, the curve is defined between x = -1 and x = 4, so those will be our limits.
Step 2: Set up the integral. The formula for the volume using cylindrical shells is V = 2π ∫[a,b] x * f(x) dx, where f(x) is the function defining the curve and [a, b] are the limits of integration.
Step 3: Calculate the integral. In our case, the integral becomes V = 2π ∫[-1, 4] x * (x^4) dx.
Step 4: Evaluate the integral. Integrate x * (x^4) with respect to x. This will give you the antiderivative of x^5 / 5.
Step 5: Substitute the limits of integration into the integral. Evaluate the antiderivative at the upper limit (4) and subtract the result of evaluating it at the lower limit (-1).
Step 6: Multiply the result by 2π to get the final volume value.
b) To find the volume obtained by revolving the area under the curve f(x)=x^4 around the line y = -5, you can use the method of disks or washers. Here are the steps:
Step 1: Determine the limits of integration. In this case, the curve is still defined between x = -1 and x = 4, so those will be our limits.
Step 2: Set up the integral. The formula for the volume using disks or washers is V = π ∫[a,b] (R^2 - r^2) dx, where R is the radius of the outer disk (distance from the curve to the line of revolution) and r is the radius of the inner disk (distance from the x-axis to the line of revolution).
Step 3: Calculate the radii. In our case, R is the distance between the curve f(x) and the line y = -5, which is f(x) + 5. r is the distance between the x-axis and the line y = -5, which is 5.
Step 4: Substitute the radii into the integral. This will give you V = π ∫[-1, 4] ((f(x) + 5)^2 - 5^2) dx.
Step 5: Calculate the integral. Expand and simplify the expression ((f(x) + 5)^2 - 5^2) and integrate it with respect to x.
Step 6: Evaluate the integral. Substitute the limits of integration into the integral and calculate the result.
Step 7: Multiply the result by π to get the final volume value.