Find the area of the region bounded by y x 2, the x-axis, and the lines x 0 and x 1. The function defined by y x 2 is continuous and nonnegative on [0, 1] so the
PROBLEM SOLUTION
1
1
§ x3 ¶ 1 area is given by ¯ x dx ¨ · square units. © 3 ¸0 3 0 2
EXERCISE
11·1
Find the area of the region bounded by the indicated curves.
1. y 2 x 2 2 x 24 ; x- axis; x 3; x 6 P 2P 2. y sin x ; x-axis; x = ; x 3 3 3. y 8 x 2 x 2 ; x-axis; x =1; x = 3 P 4. y sec2 x ; x-axis; y-axis; x = 4 5. y 4 x 4 ; x-axis; y-axis; x = 8 P 6. y cos x ; x-axis; y-axis; x = 6
Area of a region between two curves If f and g are continuous functions with f ( x ) q g ( x ) on [a, b], then the area between the two b
curves is given by ¯ [ f ( x ) g ( x )] dx . a
As you can see, the problem of finding areas between curves involves essentially exploiting ideas developed in the first section of this chapter. PROBLEM
SOLUTION
Find the area enclosed by the curves y f ( x ) x 3 x and y h( x ) sin x , the P x-axis, and the lines x and x P. 2 §P ¶ Both f and h are continuous and nonnegative in ¨ , P · and f ( x ) q h( x ) ©2 ¸ P §P ¶ , P · . Thus, the specified area is given by ¯ [( x 3 x ) sin x ] dx P /2 ©2 ¸
on ¨
P
§ x4 x2 ¶ § P 4 P 2 ¶ § P 4 P 2 ¶ ¤ 15P 4 3P 2 ³ 1´ square units. ¨ cos x · ¨ 1· ¨ 6 3 · ¥ 4 2 4 2 64 8 2 2 ¦ µ © ¸ P /2 © ¸ © ¸ PROBLEM SOLUTION
Find the area enclosed by the lines x 0, x 1, the x-axis, and the curves y f ( x ) x 1 and y g ( x ) x 2. Solve for the intersection of the two functions by equating the expressions to get x 2 x 1 or x 2 x 1 0. The solutions to this quadratic equation are x
84
5 1 1 o 5 and the value in the interval [0, 1] is x . Also, f 2 2
Applications of the Derivative and the Definite Integral