Previous exam questions on area between functions and volumes of solids.

1.Let f(x) = cos(x2) and g(x) = ex, for –1.5 ≤ x ≤ 0.5.
Find the area of the region enclosed by the graphs of f and g. (Total 6 marks)

2.Let f(x) = Aekx + 3. Part of the graph of f is shown below.

The y-intercept is at (0, 13).

(a)Show that A =10.
(2)

(b)Given that f(15) = 3.49 (correct to 3 significant figures), find the value of k. (3)

(c)(i)Using your value of k, find f′(x).
(ii)Hence, explain why f is a decreasing function.
(iii)Write down the equation of the horizontal asymptote of the graph f. (5)

Let g(x) = –x2 + 12x – 24.
(d)Find the area enclosed by the graphs of f and g.
(6)
(Total 16 marks)

3.The following diagram shows the graphs of f (x) = ln (3x – 2) + 1 and g (x) = – 4 cos (0.5x) + 2, for 1 £ x £ 10.

(a)Let A be the area of the region enclosed by the curves of f and g. (i)Find an expression for A.
(ii)Calculate the value of A.
(6)

(b)(i)Find f ′ (x).
(ii)Find g′ (x).
(4)

(c)There are two values of x for which the gradient of f is equal to the gradient of g. Find both these values of x. (4)
(Total 14 marks)

4.The graph of f(x) = , for –2 ≤ x ≤ 2, is shown below.

The region enclosed by the curve of f and the x-axis is rotated 360° about the x-axis. Find the volume of the solid formed.
(Total 6 marks)

5.The graph of y = between x = 0 and x = a is rotated 360° about the x-axis. The volume of the solid formed is 32π. Find the value of a. (Total 7 marks)

...Surface Area to Volume Ratio and the Relation to the Rate of Diffusion
Aim and Background
This is an experiment to examine how the Surface Area / Volume Ratio affects the rate of diffusion and how this relates to the size and shape of living organisms.
The surface area to volume ratio in living organisms is very important. Nutrients and oxygen need to diffuse through the cell membrane and into the cells....

...Investigating Ratios of Areas and Volumes
In this portfolio, I will be investigating the ratios of the areas and volumes formed from a curve in the form y = xn between two arbitrary parameters x = a and x = b, such that a < b. This will be done by using integration to find the area under the curve or volume of revolution about an axis. The two areas that will be compared will be labeled ‘A’...

...Mathematics
Volume of Solids
Formulae for Volume of Solids
Cube | Cuboid | Triangular Prism | Cylinder | Cone | Pyramid | Sphere | AnyPrism |
s3 | lwh | ½bhl | Πr2h | 1/3πr2h | 1/3Ah | 4/3πr3 | Ah |
A = area of the base of the figure
s = length of a side of the figure
l = length of the figure
w = width of the figure
h = height of the figure
π = 22/7 or 3.14
1. Compute the volume of a cube with side 7cm....

...Surface area / Volume ratio Experiment
Introduction:
The surface area to volume ratio in living organisms is very important. Nutrients and oxygen need to diffuse through the cell membrane and into the cells. Most cells are no longer than 1mm in diameter because small cells enable nutrients and oxygen to diffuse into the cell quickly and allow waste to diffuse out of the cell quickly. If the cells were any bigger than this then it...

...across the cell membrane and the cell. The movements within a cell are called Diffusion. When molecules move across a cell membrane it is known as Osmosis. Diffusion is the process by which molecules of a substance move from areas of higher concentration of that substance to areas of lower concentration. Diffusion can be the transfer of anything anywhere. However, that is not true for osmosis. Osmosis is diffusion, but a specific type of diffusion. Osmosis is...

...(Term–II)
13
SURFACE AREAS AND VOLUMES
A. SUMMATIVE ASSESSMENT
(c) Length of diagonal =
TH
G
(a) Lateral surface area = 4l2 (b) Total surface area = 6l2 (c) Length of diagonal = 3 l 3. Cylinder : For a cylinder of radius r and height h, we have : (a) Area of curved surface = 2πrh
BR
O
ER
2. Cube : For a cube of edge l, we have :
O
YA L
TEXTBOOK’S EXERCISE 13.1 Unless stated otherwise,...

...Ratio of
Surface Area to Volume on
Transformation Efficiency in
Escherichia coli
K-12
Eliana Pouchard, Steve Teng, and Cameron Wong
Department of Microbiology & Immunology, UBC
In the process of transformation, bact
eria take up DNA fr
om the environment
through their cell wall. To induce competence in
the cells, the DNA for uptake must first
attach to the cell surface prio
r to passing throug
h the membrane. Previous studies on the
effect...

...Background information:
1) What is diffusion?
Movement of a particular type of molecule from an area of high concentration to an area of low concentration.
2) How is diffusion used by living cells?
Living cells bring in food, water and oxygen, and excrete wastes through the process of diffusion
3) List two body systems in vertebrates that are dependent on diffusion
Digestive system and respirational system
4) What is meant by the term metabolism
the...

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