Math 152 Hardy Name Midterm Exam I 10302020

Math 152 Hardyname Midterm Exam I 10302020

Evaluate the following integrals. Given the region is made up of circular arcs and straight line segments, and understanding the geometric context implied by the problem statements, perform the necessary calculus steps by hand, justify your work, and set up the integrals as required. The questions include evaluating integrals, determining average temperature, and setting up volume integrals using both shell and washer methods.

Paper For Above instruction

The midterm examination for Math 152 in the context of rigorous calculus assessment involves multiple computational and setup tasks designed to evaluate understanding of integration, geometric interpretation, and volume calculation methods. This comprehensive exam requires detailed work, including integral evaluation, temperature averaging, and volume computation via shell and washer methods, which are fundamental techniques in multivariable calculus and mathematical modeling.

1. Evaluate the following integrals:

• ∫ (1 + x²)^{3/2} dx
• ∫₀^π/3 sin(θ) cos²(θ) dθ
• ∫₀^6 (x² - 1) dx
• ∫_{region} (x² + y²)^{5} dy

For the integral ∫ (1 + x²)^{3/2} dx, substitution u = 1 + x² leads to du = 2x dx, but since the integrand involves x², rewriting or substituting u to simplify integration by parts or recognizing kv double substitution is effective. For the second integral, utilize the substitution t = cos(θ), with dt = -sin(θ) dθ, simplifying the integral to a single variable. The third integral involves polynomial functions, straightforward to evaluate via basic antiderivatives. The fourth integral involves a region in the plane described by these functions, potentially requiring conversion to polar coordinates for simplification.

2. What is the average temperature between 9 am and 9 pm?

Assuming the temperature T(t) is modeled by the function 13 sin (π/12 t) + 55, where t is measured in hours from 9 am (t=0) to 9 pm (t=12). The average temperature is given by:

Average temperature = (1/12) ∫₀^{12} T(t) dt

Evaluating this integral involves integrating sine functions over a full period and adding the constant, leading to a straightforward calculation of the mean temperature over the specified interval.

3. Volume of the solid obtained by revolving region R around the y-axis:

• (a) Sketch: Drawing the region R and the resulting solid S provides geometric intuition, indicating the shape, boundaries, and the axis of rotation.
• (b) Shell method setup: The volume V using cylindrical shells is given by integrating the shell circumference times height over the region, expressed as:

   V = ∫_{a}^{b} 2π x * height(x) dx

  where height(x) is the difference between the functions defining the region's upper and lower boundaries.

• (c) Washer method setup: The volume V using washers is constructed from integrating the area of cross-sectional disks with holes, formulated as:

   V = π ∫_{a}^{b} [R_{outer}^2 - R_{inner}^2] dx

  where R_outer and R_inner are the distances from the y-axis to the outer and inner boundaries of the region respectively.

These steps and methods symbolize core techniques in calculus, blending algebraic manipulation, integral calculus, and geometric visualization to accurately analyze and compute problems involving areas, averages, and volumes.

References

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