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Chapter 4: Q. 1 (page 327)

Given formula for the areas of each of the following geometric figures
a) area of circle with radius r
b) a semicircle of radius r
c) a right triangle with legs of lengths a and b
d) a triangle with base b and altitude h
e) a rectangle with sides of lengths w and l
f) a trapezoid with width w and height $h_{1}, h_{2}$

Short Answer

Expert verified

The areas can be given as

$a) a r e a o f c i r c l e w i t h r a d i u s r = π r^{2} b) a s e m i c i r c l e o f r a d i u s r = \frac{π r^{2}}{2} c) a r i g h t t r i a n g l e w i t h l e g s o f l e n g t h s a a n d b = \frac{1}{2} \cdot a \cdot b d) a t r i a n g l e w i t h b a s e b a n d a l t i t u d e h = \frac{1}{2} \cdot b \cdot h e) a r e c \tan g l e w i t h s i d e s o f l e n g t h s w a n d l = w \cdot l f) a t r a p e z o i d w i t h w i d t h w a n d h e i g h h_{1}, h_{2} = \frac{1}{2} \cdot w (h_{1} + h_{2})$

Step by step solution

01

Given information

We are given geometric figures we have to find their areas

02

Give the formulas

We have,

$a) a r e a o f c i r c l e w i t h r a d i u s r = π r^{2} b) a s e m i c i r c l e o f r a d i u s r = \frac{π r^{2}}{2} c) a r i g h t t r i a n g l e w i t h l e g s o f l e n g t h s a a n d b = \frac{1}{2} \cdot a \cdot b d) a t r i a n g l e w i t h b a s e b a n d a l t i t u d e h = \frac{1}{2} \cdot b \cdot h e) a r e c \tan g l e w i t h s i d e s o f l e n g t h s w a n d l = w \cdot l f) a t r a p e z o i d w i t h w i d t h w a n d h e i g h h_{1}, h_{2} = \frac{1}{2} \cdot w (h_{1} + h_{2})$

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Most popular questions from this chapter

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. $\int (x^{2} - 3 x^{5} - 7) d x$

Sum and constant-multiple rules: State the sum and constant-multiple rules for (a) derivatives and (b) limits.

Compare the definitions of the definite and indefinite integrals. List at least three things that are different about these mathematical objects.

Show by exhibiting a counterexample that, in general, $\int f (x) g (x) d x \neq (\int f (x) d x) (\int g (x) d x)$ . In other words, find two functions f and g so that the integral of their product is not equal to the product of their integrals.

Determine which of the limit of sums in Exercises 47–52 are infinite and which are finite. For each limit of sums that is finite, compute its value

$\lim_{n \to \infty} \sum_{k = 1}^{n} \frac{k^{3}}{n^{4} + n + 1}$

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