The figure shows the region bounded by the x-axis and the graph of . Use Formulas (42) and (43)-which are derived by integration by parts-to find (a) the area of this region; (b) the volume obtained by revolving this region around the y-axis.
To find the area of the region bounded by the x-axis and the graph of a function, you can use the formula for calculating the definite integral. In this case, let's assume the function is denoted by f(x).
(a) Area of the region:
To calculate the area, you need to evaluate the definite integral of the function f(x) over the interval where the graph lies above the x-axis. You can use the formula for the definite integral as follows:
Area = ∫[a, b] f(x) dx,
where [a, b] is the interval over which the graph lies above the x-axis.
(b) Volume obtained by revolving the region around the y-axis:
To calculate the volume obtained by revolving the region around the y-axis, you can use the method of cylindrical shells. This method involves integrating the product of the circumference of each cylindrical shell and its height.
The height of each cylindrical shell is determined by the function f(x), and the circumference is determined by the differential element dx. The formula for calculating the volume using cylindrical shells is:
Volume = 2π ∫[a, b] x * f(x) dx,
where [a, b] is the interval over which the graph lies above the x-axis.
To find the values of a and b, you may need to analyze the given formulation or problem statement to determine the appropriate limits of integration.
Please note that formulas (42) and (43), which are derived by integration by parts, may not be directly relevant to these specific calculations. It is important to use the appropriate formulas for the specific calculations you are trying to perform.