The diagram shows a shaded region, bounded by the curves y = 2 x , y = - 2 , x = 1 and x = 4 . The area of the region can be written in the form p ln ( 2 ) + q . Find the values of p and of q .

Given curves are: y = 2 x , y = - 2 , x = 1 Now, the definite integral for the area between the two functions is A = ∫ x 1 x 2 g x - f x d x i.e., A = ∫ 1 4 2 x - ( - 2 ) d x Therefore, A = ∫ 1 4 2 x + 2 d x ⇒ A = 2 ln x + 2 x 1 4 ⇒ A = 2 ln 4 + 2 4 - 2 ln 1 + 2 1 ⇒ A = 2 ln 4 + 8 - 2 ⇒ A = 2 ln 2 2 + 6 ⇒ A = 4 ln 2 + 6 On comparing with the given expression p ln 2 + q , we have: p = 4 , q = 6 .

Consider the functions g x = 1 2 x and f x = x . Part of the graphs of f and g are shown in the diagram. Find the shaded region area.

Given functions: g x = 1 2 x and f x = x Finding the x - coordinates of the points of intersection of the two functions. Now, the point of intersection of two curves can be calculated by equating f x = g x i.e., 1 2 x = x Therefore, x = 2 x ⇒ x 2 = 4 x [Squaring on both sides] ⇒ x 2 - 4 x = 0 ⇒ x x - 4 = 0 ⇒ x = 0 or 4 Now, if x = 0 , then we have y = 0 and similarly, if x = 4 , then we have y = 2 Hence, the coordinates of the point of intersection of graphs of f and g are: ( 0 , 0 ) , ( 4 , 2 ) . Now, the definite integral for the area between the two functions is A = ∫ x 1 x 2 ( f ( x ) - g ( x ) ) d x ⇒ A = ∫ 0 4 x - 1 2 x d x ⇒ A = ∫ 0 4 x 1 2 - 1 2 x d x ⇒ A = x 1 2 + 1 1 2 + 1 - 1 2 × x 1 + 1 1 + 1 0 4 ⇒ A = 2 3 x 3 2 - 1 4 x 2 0 4 ⇒ A = 2 3 4 3 2 - 1 4 4 2 - 0 ⇒ A = 16 3 - 4 ⇒ A = 4 3 sq.units Hence, the required area of the shaded reign is 4 3 sq.units.

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Sketch the graph of the given functions on the same set of axes, and shade the region bound by the two curves. Write down an expression that gives the area of the region. Find the area using a GDC.

$y = \ln x and y = \frac{1}{2} x - 2$ . [Use, $\ln (8.21093) = 2.10546619354, \ln (0.145351) = - 1.92860377206$ ]

Important Questions on From Approximation to Generalization: Integration

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Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 10 - From Approximation to Generalization: Integration>Exercise 10J>Q 6

Sketch the graph of the given functions on the same set of axes, and shade the region bound by the two curves. Write down an expression that gives the area of the region. Find the area using a GDC.

$f (x) = x^{2} - 5 x + 1 and g (x) = 6 - x^{2}$ .

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Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 10 - From Approximation to Generalization: Integration>Exercise 10J>Q 7

Sketch the graph of the given functions on the same set of axes, and shade the region bound by the two curves. Write down an expression that gives the area of the region. Find the area using a GDC.

$f (x) = 2 e^{x} and g (x) = 3 x + 4$ . [Use, $e = 2.718$ ]

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Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 10 - From Approximation to Generalization: Integration>Exercise 10J>Q 8

Sketch the graph of the given functions on the same set of axes, and shade the region bound by the two curves. Write down an expression that gives the area of the region. Find the area using a GDC.

$y = \frac{x + 4}{x - 2} and y = - x - 7$ .

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Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 10 - From Approximation to Generalization: Integration>Exercise 10J>Q 9

The diagram shows a shaded region, bounded by the curves $y = \frac{2}{x}, y = - 2, x = 1$ and $x = 4$ . The area of the region can be written in the form $p \ln (2) + q$ . Find the values of $p and of q$ .

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Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 10 - From Approximation to Generalization: Integration>Exercise 10J>Q 10

Consider the functions $g (x) = \frac{1}{2} x and f (x) = \sqrt{x}$ . Part of the graphs of $f and g$ are shown in the diagram. Find the coordinates of the points of intersection of the graphs of $f and g$ .

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Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 10 - From Approximation to Generalization: Integration>Exercise 10J>Q 10

Consider the functions $g (x) = \frac{1}{2} x and f (x) = \sqrt{x}$ . Part of the graphs of $f and g$ are shown in the diagram. Write an integral expression for the area enclosed by the graph of $f and g$ .

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Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 10 - From Approximation to Generalization: Integration>Exercise 10J>Q 10

Consider the functions $g (x) = \frac{1}{2} x and f (x) = \sqrt{x}$ . Part of the graphs of $f and g$ are shown in the diagram. Find the shaded region area.

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Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 10 - From Approximation to Generalization: Integration>Exercise 10J>Q 10

Consider the functions $g (x) = \frac{1}{2} x and f (x) = \sqrt{x}$ . Part of the graphs of $f and g$ are shown in the diagram. The line $x = k$ divides the area of the region in half. Write down an integral expression for half the area of the region.

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Sketch the graph of the given functions on the same set of axes, and shade the region bound by the two curves. Write down an expression that gives the area of the region. Find the area using a GDC. y=lnx and y=12x-2. [Use, ln(8.21093)=2.10546619354,ln(0.145351)=-1.92860377206]

Important Questions on From Approximation to Generalization: Integration

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