5 Let Rbedererion in the fest quadrant enclosed by the graphs of y and y r as sh

Question

5 Let Rbedererion in the fest quadrant enclosed by the graphs of y and y r as shown below. a) Find the area of 41 The region R is the base of asolid. For this solid, at each the cross section pependicular to the avis has area x) sin(tx Find the volume of the solid not Another solid has the same base R. For this solid, the cross sections perpendicular evaluate, an integral expression forthe volume ofthe solid. are squares. Write, but do to the y do Write but do not evaluate the integral which gives the volume of the solid formed by routing Raround the line y-5 e) Write but do not evaluate, the integral which gives the volume of the solid formed by rotating Ranound the line r --1 6 [Calculator] Let R be the region bounded by graphs of y sin(ra) and y r'-4r, as shown in the figure above. a Find the area of R. b The horizontal line y"-2 splits the region R into two parts. Write, but do not evaluate, an integral expression for the area ofthe pantofR that is below this horizontal is given ne region R the At l points in Rata distance x from the y-avis, the depth of the water by h (x)-3 Find the volume of water in the pond.

Explanation / Answer

Region R is the area under y1= 2x and y1= x2 from x =0 to x =2

Hence area = Integral 0 to 2 of (2x-x2) dx

= x2-x3/3

= (4) -1/3(8)

= 4/3 square units.

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Volume = area x height here height = dx

Hence volume= integral 0 to2 of sin x/2

= -2 cos x/2

Substitute 0 and 2

= -2 cos 1 + 2 cos 0

= 2+2 = 4 cubic units

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Here pependicular to y axis.

Hence volume = integral A(y) dy where 0<y<4

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