Integration

Questions & Answers

The base of a solid is the region in the xy-plane bounded by 4 2 -² and y = 0. Every cross-section of the curves y = 4 - 25 the solid parallel to the x-axis is a triangle whose height and base are equal. The volume of this solid is:


A hole in the ground in the shape of an inverted cone is 22 meters deep and has radius at the top of 18 meters. This cone is filled to the top with sawdust. The density, p, of the sawdust in the hole depends upon its depth, x: p(x) = 2.7 - 1.4e-0.32 kg m³ The total mass of sawdust in the conical hole is: kg


Find the volume formed by rotating about the y-axis the region enclosed by: 3 6y and y³ = x with y ≥ 0


The point of this question is to make sure you fully understand the logic behind volumes by slicing (and aren't just plugging it into a formula that doesn't really make any sense to you.) The way to fully understand them is to understand the picture of a slice. The base of a solid is the region in the xy-plane between the the lines y = 0, y = √√x, x= 1 and 2 = 3125. = Cross-sections of the solid perpendicular to the x-axis (and to the xy-plane) are semicircles whose diameter is on the base. (So be careful with the radius of your slices.) To get full credit for this question, please do the following: 1. Draw a picture of the full base, labeling all relevant points. (This might include needing to do some algebra to determine those relevant points. Please NEATLY show that algebra work.) 2. In that base that you drew in part 1, draw a rectangle that would represent the base of one general slice and label its width and length with appropriate variable(s). 3. Draw the picture of the slice determined by that rectangle you drew in part 2. Then indicate clearly what the volume of that slice would be (In other words, what would AV be for that slice?) Note: this volume would include variable(s). 4. Lastly, write down the integral that represents the volume of the full solid and evaluate it with technology.


(1) (2 marks) Find the volume of the solid enclosed by the surface


(3) (4 marks) Use polar coordinates to evaluate the following integrals:


To find the blue shaded area above, we would calculate:


Speedometer readings for a vehicle (in motion) at 8-second intervals are given in the table. Estimate the distance traveled by the vehicle during this 40-second period using the velocities at the beginning of the time intervals.


Liquid leaked from a damaged tank at a rate of r(t) liters per hour. The rate decreased as time passed and values of the rate at two-hour time intervals are shown in the table.


Evaluate the definite integral by interpreting it in terms of areas.


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