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**Q1:**(5) (5 marks) Let H be a solid hemisphere of radius a with constant density. (a) Find the centroid of H. (b) Find the moment of inertia of H about a diameter of its base.See Answer**Q2:**Using the trapezium rule, we can calculate the integral of a function, f(x), between the limits of x=a and x=b to be approximately : \int_{a}^{b} f(x) d x=\frac{h}{2}\left[f(a)+f(b)+2 \sum_{i=1}^{n-1} f(a+i h)\right] \text { Write a program to calculate the integral of } \sin ^{2}(x) \text {. } Your program should : Ask the user for the values of a, b and how many points between these values. Use the inline command in the script to evaluate sin²(x) when calculating the integral. \text { - Be tested using the information that } \int_{0}^{\pi} \sin ^{2}(x) d x=\frac{\pi}{2} The error function, erf(z), is used in solving the heat equation. It is given by : \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-x^{2}} d x Write a program which will calculate the value of the erf(z), adapted from your trapezium integrationprogram above. You can test you program using the information that erf(1) = 1.See Answer**Q3:**Explain why the areas for each of the following graphs are equal WITHOUT computing their u/du-substitution. It may be helpful to recall that sin(2x) = 2 sin x cosx. areas. Hint: UseSee Answer**Q4:**Using full sentences, write a brief summary of the geometric transformation that occurs to the integralSee Answer**Q5:**Luca and Maria are debating about how to evaluate the integralSee Answer**Q6:**The goal of this problem is to show the following integral formula holds.See Answer**Q7:**Evaluate the following integrale.See Answer**Q8:**Ed, Jin, and Faye were debating on how to integrate the following the integral:See Answer**Q9:**The goal of this problem is to find a general formula for the area of an ellipse in terms of the lengths of its semi-major and semi-minor axes (denoted by a and b in the figure below).See Answer**Q10:**Given the rational function 1/(x-1)²(x-2) Show there are no real numbers A and B that make the following partial fraction decomposition true:See Answer**Q11:**Explain your strategy for evaluating the following indefinite integral. Be sure to explain the reasoning for your strategy and any substitution choices that would be needed to show the following statement is true.See Answer**Q12:**Identify which Jamal's method(s), if any, you would use to integrate the following integrals. Be sure to provide reasoning for your choices.See Answer**Q13:**4 1. f(x) = 3x³ -1/2 x² 3 2. g(x) = (4-¹x²)³See Answer**Q14:**3. h(x) = sin(x) 4. k(x) = ln(3x² + 2)See Answer**Q15:**5. f(x) = x² cos(x) + x tan(x) 6. g(x) = √3x²+2See Answer**Q16:**7. h(x) = = =sin-¹(x) 8. Using derivative rules and the trigonometric identities 1 sin² (8) = (1 - cos(20)), cos² (0) = (1 + cos(20)), show clearly that the function f(x) = sin(x) cos(x) — satisfies properties (a) and (b). 2 (a) f'(x) = cos(2x), (b) ƒ ( 7 ) = 0See Answer**Q17:**Prove the following equalities and inequalities: $Pdv # 0, $Tds #0 $d(PV) = 0, $d(TS) = 0 $Pdv=-SAT $VdP=-$TdS $VdP = $SATSee Answer**Q18:**9.1 Add it up (#integration) For each of the following sums: (i) Estimate the sum for n= 10, 100 and 1000 (you may use computational tools to help, make sure to include supporting code). (ii) Evaluate (or estimate) the limit as n → ∞. (iii) Rewrite the sum as a definite integral and compute it. Compare your results. 2n (a) › Σ ((¹ + :-)²) = (b) Σ (COS ()) 2nSee Answer**Q19:**The base of a solid is the region in the xy-plane between the the lines y = x, y = 6x, x = 1 and x = 3. Cross-sections of the solid perpendicular to the x-axis (and to the xy-plane) are squares. The volume of this solid is:See Answer**Q20:**The base of a solid is the region in the xy-plane bounded by the curves * = - y² + 4y + 96 and x = y² − 22y + 140. Every cross-section of this solid perpendicular to the y-axis (and to the xy-plane) is a half-disk with the diameter of the half-disk sitting in the xy-plane. The volume of this solid is:See Answer

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