Piecewise Integration Involving the Floor Function
Consider the function \(f(x) = \lfloor x \rfloor\) defined on the interval \([2,7]\). (a) Analyze the function to determine the subintervals on which \(\lfloor x \rfloor\) remains constant and state the constant value on each interval. (b) Using your analysis from part (a), evaluate the definite integral \(\int_{2}^{7} \lfloor x \rfloor\,dx\) by expressing it as […]
Piecewise Integration for Area Calculation
A vehicle decelerates along a straight road following a piecewise speed function due to varying braking forces. The speed function is defined differently over two time intervals. Overall, determine the area under the speed-time curve to find the total distance traveled during deceleration. Proceed as follows: (a) Set up the piecewise function for the deceleration […]
Expected Value and Variance from a Piecewise PDF
A continuous random variable \(X\) has a piecewise probability density function defined by: \( f(x) = \begin{cases} c\,x & \text{for } 0 \le x < 2, \\ c\,(8-x) & \text{for } 2 \le x \le 4, \\ 0 & \text{otherwise} \end{cases} \) (a) Determine the normalizing constant \(c\). (b) Calculate the expected value \(E(X)\) of […]
Ensuring Velocity Continuity in Piecewise Accelerations
An object moves along a straight line with a piecewise-defined acceleration function. For \(t < 4\) seconds, the acceleration is given by \(a_1(t) = 3t\). For \(t \ge 4\) seconds, the acceleration switches to \(a_2(t) = 5 – 2t\). Given that the velocity function \(v(t)\) is continuous at \(t = 4\) seconds, determine the velocity […]
Volume of Revolution Using the Washer Method for a Piecewise Function
Consider the region bounded by the piecewise function defined as follows: – For \(0 \le x \le 2\): \(y = \sqrt{4 – (x-2)^2}\) – For \(2 < x \le 4\): \(y = \sqrt{4 – (x-4)^2}\) These functions represent the upper halves of two circles of radius 2 (one centered at \((2,0)\) and the other at […]
Volume of Revolution with Piecewise Parameter Ranges
Consider the parametric curve defined by: For \(0 \le t \le 2\): \(x(t) = t^2\) and \(y(t) = 3t\). For \(2 < t \le 4\): \(x(t) = 2t + 1\) and \(y(t) = 4t – 2\). Set up the integral expression to calculate the volume when this curve is revolved about the x-axis. Clearly indicate […]
Part b
Set up and evaluate the volume of the solid formed when the region is revolved about the y-axis.
Indefinite Integration of a Piecewise Linear Function
Evaluate the indefinite integral \(\int (3 – 2|x-1|)\,dx\) given that the function inside the absolute value, \(x-1\), changes sign at \(x = 1\).
Part b
Using the antiderivative F(x) determined in part (a), evaluate the definite integral \(\int_{0}^{3} f(x)\,dx\), where f(x) is defined as above.