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7339 quizzes
A farmer in Kenya sketches the edge of a field as the curve y = 16 - x² and covers the region under it with four rectangles of width 1 metre, as in the figure. The top right corner of each rectangle touches the curve, and the last rectangle has height 0. What is the total area of the four rectangles, in square metres? New
High School (11–12) · Application of Integrals
APA-LevelIBAPA-LevelIB
StemPy LabStemPy Lab
Easy
A textbook question reads: find the area bounded by the curve y = x², the x-axis and the ordinates x = 1 and x = 3. A student who has studied in another country has not met the word ordinate. What do the ordinates x = 1 and x = 3 mean in this question? New
High School (11–12) · Application of Integrals
APA-LevelIBAPA-LevelIB
StemPy LabStemPy Lab
Easy
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 circular pond in a park in Delhi has the equation x² + y² = 100, where lengths are in metres. A straight bridge along the x-axis divides the pond into two equal halves. What is the area of the upper half of the pond, in square metres? New
High School (11–12) · Application of Integrals
APA-LevelIBAPA-LevelIB
StemPy LabStemPy Lab
Easy
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
Two curves meet at x = 1 and at x = 4. Between these two values the curve y = p(x) lies above the curve y = q(x). Which integral gives the area of the region enclosed between the two curves? New
High School (11–12) · Application of Integrals
APA-LevelIBAPA-LevelIB
StemPy LabStemPy Lab
Easy
The figure shows the parabola y = x² and the line y = 2x + 3. The line cuts the parabola at the two marked points. What is the area of the shaded region between them? New
High School (11–12) · Application of Integrals
APA-LevelIBAPA-LevelIB
StemPy LabStemPy Lab
Easy
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
A designer models a cone-shaped lampshade by turning the straight line y = 2x, for 0 ≤ x ≤ 3, about the x-axis. Here x and y are in centimetres. What is the volume of the solid? New
High School (11–12) · Application of Integrals
APA-LevelIBAPA-LevelIB
StemPy LabStemPy Lab
Easy
A solid ball is made by turning the half circle y = √(36 - x²), for -6 ≤ x ≤ 6, about the x-axis. The lengths are in centimetres. What is the volume of the ball? New
High School (11–12) · Application of Integrals
APA-LevelIBAPA-LevelIB
StemPy LabStemPy Lab
Easy
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
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