# 4Q application of Definite integral?

1.A bowl is a solid of revolution bounded by the surfaces obtained by rotating the curves (C1) x^2=4y and (C2) x^2=8(y-k) about the y-axis, where k is a constant greater than zero. The curves are shown in Figure 11, with the part which generates the solid shaded. (i)what is the diameter of the mouth of the bowl, in terms of k? (ii)find, in terms of π and k, the capacity of the bowl (iii) water is poured into the bowl so that its greatest depth is k/2. What fraction of the capacity of the bowl is filled? (iv) show that for all positive vlues of k, the volume of the material used in making the bowl is equal to the capacity of the bowl.

2.A(4,6) is a point on the curve y^2=9x. (i) find the coordinates of the point P on the curve such that the tangent to the curve at P is parallel to the line AO, where O is the origin (ii)denote by R the region bounded by the curve and the line OA. Prove that the area of the region R is 4/3 of that of the triangle OAP (iii)find the volume of the solid generated by revolving the region R about the x-axis

3.Given the parabola C: x^2=8y, and the line L: 2y=3x-8 (a)find the points of intersection of C and L (b)find the area of the shaded portion bounded by y=0, x=8, C and L (c)find the volume generated by the area in (b) about the x-axis

4.(a) by using a definite integral, show that the volume of a right circular cone of height h, whose base is a circle of radius r is 1/3πr^2 h. (b) a container in the shape of a right circular cone is formed by removing a sector from a circular disc of radius R and joining the straight edges of the remaining portion. For a fixed R, find the maximum capacity of the container

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ans

1.(i)4√2k(ii)4πk^2(iii)1/4

3.(a)(4,2),(8,8) (b)20 (c)4544π/45

4.(b)(2√3πR^3)/27

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Question：

1. A bowl is a solid of revolution bounded by the surfaces obtained by rotating the curves C₁：x² = 4y and

C₂：x² = 8(y - k) about the y-axis, where k is a constant greater than zero. The curves are shown in

Figure 11, with the part which generates the solid shaded.

(i) What is the diameter of the mouth of the bowl, in terms of k?

(ii) Find, in terms of π and k, the capacity of the bowl

(iii) Water is poured into the bowl so that its greatest depth is k/2. What fraction of the capacity of the bowl is

filled?

(iv) Show that for all positive vlues of k, the volume of the material used in making the bowl is equal to the

capacity of the bowl.

2. A(4,6) is a point on the curve y² = 9x.

(i) Find the coordinates of the point P on the curve such that the tangent to the curve at P is parallel to the

line AO, where O is the origin

(ii) Denote by R the region bounded by the curve and the line OA. Prove that the area of the region R is 4/3 of

that of the triangle OAP

(iii) Find the volume of the solid generated by revolving the region R about the x-axis

3.Given the parabola C: x² = 8y, and the line L: 2y = 3x - 8

(a) Find the points of intersection of C and L

(b) Find the area of the shaded portion bounded by y = 0, x = 8, C and L

(c) Find the volume generated by the area in (b) about the x-axis

4.(a) By using a definite integral, show that the volume of a right circular cone of height h, whose base is a circle

of radius r is πr²h/3.

4.(b) A container in the shape of a right circular cone is formed by removing a sector from a circular disc of radius R and joining the straight edges of the remaining portion. For a fixed R, find the maximum capacity of the

container

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20160911

Solution：