Unit 3: Integrals

Answer: C — The correct answer is C) 23 cubic meters. To find the total amount of water, we need to calculate the area under the rate curve, which can be broken down into three rectangles: (2 cubic meters/minute * 5 minutes) + (3 cubic meters/minute * 3 minutes) + (1 cubic meter/minute * 2 minutes) = 10 + 9 + 4 = 23 cubic meters. Option A is incorrect because it underestimates the amount of water. Option B is incorrect because it overestimates the amount of water for the first 5 minutes. Option D is incorrect because it overestimates the amount of water for all time intervals.

Answer: B — The correct answer is B because the definite integral of a function f(x) from a to b represents the area between the curve of f(x) and the x-axis from a to b. Since f(x) = 2x^2 + 1 is above the x-axis for all x in [0,2], the definite integral represents the area between the curve and the x-axis. Option A is incorrect because the area below the x-axis would be represented by a negative definite integral, but f(x) is above the x-axis. Option C is incorrect because f(x) is above the x-axis, so the definite integral is positive. Option D is incorrect because the definite integral is a number, whereas the indefinite integral is a function.

Answer: A — The correct answer is A because the displacement of the ball is given by the definite integral of the velocity function with respect to time, which is ∫[0,2] (20 - 9.8t) dt. This integral is evaluated as 20t - 4.9t^2 from 0 to 2, which equals 20.4 meters. Option B is incorrect because it subtracts the value of the integral at the upper limit from the value at the lower limit, which is the opposite of the correct procedure. Option C is incorrect because it neglects to subtract the value of the integral at the lower limit, and also incorrectly states that the displacement is 0 because the ball goes up and comes back down. Option D is incorrect because it only evaluates the integral at the lower limit and does not find the definite integral.

Answer: A — The correct answer is A) 20.41 meters. To find the total distance traveled by the ball, we need to integrate the absolute value of the velocity function with respect to time, from the time the ball is thrown (t = 0) until it reaches its maximum height. The ball reaches its maximum height when its velocity is zero, so we need to find the time at which v(t) = 0. Solving 20 - 9.8t = 0 gives t = 2.04 seconds. The distance traveled is the integral of |v(t)| from 0 to 2.04, which is equal to the integral of v(t) from 0 to 2.04 since v(t) is positive in this interval. The integral of v(t) = 20 - 9.8t from 0 to 2.04 is [20t - 4.9t^2] from 0 to 2.04, which equals 20(2.04) - 4.9(2.04)^2 = 40.8 - 20.39 = 20.41 meters. The other options are incorrect because they do not represent the correct calculation of the definite integral of the velocity function over the specified time interval.

Answer: A — The correct answer is A) 25.13 minutes. To find the time to fill the tank, we first need to calculate the volume of the tank. The volume of a cylinder (which is the shape of the tank) is given by V = πr^2h, where r is the radius of the base and h is the height. Substituting the given values, we get V = π * (4)^2 * 10 = 160π ≈ 502.65 cubic meters. The tank is being filled at a rate of 2 cubic meters per minute, so the time it will take to fill the tank is the total volume divided by the rate: time = volume / rate = 502.65 / 2 ≈ 25.13 minutes. The other options are incorrect because they do not accurately reflect the calculation based on the volume of the tank and the fill rate.