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513 quizzes
The side of a ramp in a skatepark in Perth is cut along the curve y = 6√x, where x and y are in decimetres, between x = 1 and x = 4. The bottom edge of the side is along the x-axis. A painter needs the area of the side. What is the area, in square decimetres? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
On a map, the western edge of a woodland is the y-axis and the eastern edge is the curve x = 2√y. The woodland lies between the lines y = 1 and y = 9, as in the figure. Lengths on the map are in hundreds of metres. What is the area of the woodland, in square units of the map? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
A flat metal plate in a workshop in Chennai is cut to the shape of the curve 9x² + 4y² = 36, where x and y are in centimetres. The workshop needs the area of the plate to cost the metal. What is the area of the plate, in square centimetres? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
The figure shows the profile of a trench dug across a road. It follows the curve y = x² - 1 for -2 ≤ x ≤ 2, with x and y in metres. The road crew wants the total area between the profile and the line y = 0, counting the parts above and below the line as positive. What is this total area, in square metres? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
A student in Seoul estimates the area under y = x² from x = 0 to x = 1 with 10 strips of equal width. She takes the height of each strip at its left end and adds the strip areas. What does she get? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Hard
A cycle path in a nature reserve has the profile y = (x - 1)(x - 2)(x - 3) for 1 ≤ x ≤ 3, with x in kilometres and y in kilometres, as in the figure. The path crosses a reference line y = 0 at three points. The reserve wants the total area between the profile and the reference line. What is this area, in square kilometres? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Hard
A student plots the curve y = 1/(1 + x²) and wants the exact area under it from x = 0 to x = 1, with the x-axis below. Lengths are in centimetres. What is the area, in square centimetres? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
The parabola y = x² - 5x and the line y = 2x - 10 meet at two points and enclose a finite region. What is the area of this region? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
The figure shows the curves y = x² and y = x³ for 0 ≤ x ≤ 1. They meet at the origin and at (1, 1). What is the area of the shaded region between them? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
A trough has a flat lid along the line y = -2 and a curved floor along y = x² - 11, with lengths in metres, as in the figure. The cross-section is the shaded region between the lid and the floor. What is its area? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
The figure shows y = eˣ and y = e⁻ˣ from x = 0 to the dashed line x = 1. The curves start together at (0, 1). What is the area of the shaded region, to two decimal places? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
The graph of y = x³ and the line y = 16x enclose two regions, as in the figure. What is the total area of the two regions? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
The parabolas x = y² and x = 2 - y² enclose the shaded region in the figure. What is its area? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
A circular pond has radius 5 m. A straight wall crosses the pond at a distance of 3 m from its centre and divides the pond into two parts. What is the area of the smaller part, to two decimal places? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Hard
The figure shows the parabola y = x² and the roof-shaped graph y = 3 - 2|x|. What is the area of the shaded region between them? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Hard
The shaded region in the picture lies between the line y = 2x and the parabola y = x². A potter turns it about the x-axis to make a solid with a hollow inside, with x and y in centimetres. What is the volume of the solid? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
The shaded region in the picture lies between the y-axis and the curve y = 2x², from y = 0 up to y = 8. It is turned about the y-axis to make a bowl, with the lengths in centimetres. What is the volume of the bowl? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
The voltage across a sensor is f(t) = t² volts, where t is the time in seconds, for 1 ≤ t ≤ 4. A data logger reports the mean value of the voltage over these three seconds. What does it report? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
The picture shows the velocity of a trolley on a straight track, made of straight lines. It moves forward, then reverses, and then keeps reversing at a steady speed. What is the total distance it travels in the 7 seconds? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
A farm tank in a dry region is filled by a pump. The rate of inflow is r(t) = 24 - 3t litres per minute, for 0 ≤ t ≤ 8 minutes, where t is the time since the pump started. How many litres enter the tank in the eight minutes? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
A winding footpath follows the curve y = 2/3x^(3/2) from x = 0 to x = 8, with x and y in metres. The council wants to know the length of the path. How long is it? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Hard
A cyclist rides for 6 hours. Her speed v(t) in km per hour is a continuous function of the time t in hours, and she covers 30 km in all. Which statement must be true? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
A hemispherical bowl of radius 10 cm stands with its flat rim on top, as in the picture. It holds water 4 cm deep at the middle. What volume of water does it hold? New
High School (11–12) · Application of Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Hard
A cost model gives the marginal cost 3q² + 1/q for q > 0 units made. The total cost, apart from its constant, is the antiderivative of the marginal cost. Which expression is it? New
High School (11–12) · Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
The gradient of a footpath at a horizontal distance x metres from a gate is 2x + 1. The path is 3 metres above the field at x = 1 metre. Which formula gives the height y metres of the path above the field? New
High School (11–12) · Integrals
APA-LevelIB+20APA-LevelIB
StemPy LabStemPy Lab
Medium
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