Calculus Question

Math Evaluation – Integrals Questions Part 1 1. Evaluate the following integrals, without using tables and also simplifying to the minimum expression. 2. Given 𝑑𝑦 = 𝑥√100 − 𝑥 2 𝑑𝑥, 𝑦 = 0 when 𝑥 = 0. Find the value of 𝑦 when 𝑥 = 8 and 3 𝑥 = − 2 (without approximating). For the achievement of Part 1 it is required to correctly solve these two exercises. Part 2 3. Determine the values of P for which the integral converges and evaluate the integral for those values of P. 4. Determine whether the given integrals are convergent or divergent. For the achievement of Part 2 it is required to correctly solve these two exercises. Part 3 5. Find the area of the surface bounded by the hyperbola 𝑥 2 − 𝑦 2 = 𝑎2 and the line 𝑥 = 2𝑎. Without using tables. 2 2 6. Without using tables, determine the arc length of the curve 𝑥 3 + 𝑦 3 = 1 in the first 1 quadrant from the point where 𝑥 = 8 to the point where 𝑥 = 1. For the achievement of Part 3 it is required to correctly solve these two exercises. Part 4 7. Determine the area of the region inside the cardioid 𝑟 = 𝑎(1 + 𝑐𝑜𝑠𝜃) and lying outside the circumference 𝑟 = 2𝑎𝑐𝑜𝑠𝜃. 8. Be the curve: Find: 𝑑2 𝑦 a. 𝑑𝑥 2 b. The parameter values for which the curve is concave upward and concave downward. c. Inflection points. For the achievement of Part 4 it is required to correctly solve these two exercises.