Question

Estimate the area under the curve f(x) = x2 from x = 1 to x = 5 by using four inscribed (under the curve) rectangles. Answer to the nearest integer.

Answer 1

Answer:

Area under the curve

 f(x)=x², from x=1 to x=5 will be equal to

        $=\int\limits^5_1 {x^2} \, dx \\\\=[\frac{x^3}{3}]\left \{ {{x=5} \atop {x=1}} \right.\\\\=\frac{5^3}{3}-\frac{1^3}{3}\\\\ =\frac{125}{3}-\frac{1}{3}\\\\=\frac{124}{3}$

Area of the curve to the nearest integer=41 square units

⇒Using Rectangles Only

Area of Rectangle=Length × Breadth

→x=1, gives, y=1

→x=2, gives, y=4

Area of First rectangle =1×4=4 Square unit

→x=3, gives, y=9

Area of Second rectangle =1×9=9 Square unit

→x=4, gives, y=16

→x=5, gives, y=25

Area of Third rectangle =1×16=16 Square unit

Area of fourth rectangle =1×25=25  Square unit

Area of Rectangles=4+9+16+25

            =54 square units

Area of Triangle

        $=\frac{1}{2} \times \text{base} \times \text{Height}\\\\A_{1},A_{2},A_{3},A_{4},\text{are four right triangles}\\\\A_{1}=\frac{1}{2} \times1 \times 3\\\\=\frac{3}{2}\\\\A_{2}=\frac{1}{2} \times 1 \times 5\\\\=\frac{5}{2}\\\\A_{3}=\frac{1}{2} \times 1 \times 7\\\\=\frac{7}{2}\\\\A_{4}=\frac{1}{2} \times 1 \times 9\\\\=\frac{9}{2}\\\\A_{1}+A_{2}+A_{3}+A_{4}=\frac{3}{2}+\frac{5}{2}+\frac{7}{2}+\frac{9}{2}\\\\=12$    

Required Area of the Region

                      = Area of four Rectangles -Area of Triangle

                       =54 Square unit - 12 Square unit

                       =42 Square Unit

Answer 2

Check the picture below.