Find the area bounded by y = 4x + 3 with x- axis between the lines x = 1 and x = 4

Find the area of the parabola \({ y }^{ 2 }=8x\) bounded by its latus rectum.

Sketch the graph \(y=\left| x+3 \right| \) and evaluate \(\int _{ -6 }^{ 0 }{ \left| x+3 \right| } \) dx.

Using integration find the area of the circle whose center is at the origin and the radius is a units.

Using integration find the area of the region bounded between the line x = 4 and the parabola y² = 16x.

Calculate the area bounded by the parabola y² = 4ax and its latus rectum.

Find the area bounded by the curve y = x² and the line y = 4

The rate of change of sales of a company after an advertisement campaign is represented as, f (t) = 3000e^−0.3t where t represents the number of months after the advertisement. Find out the total cumulative sales after 4 months and the sales during the fifth month. Also find out the total sales due to the advertisement campaign [e^-1.2 = 0.3012, e^-1.5 = 0.2231].

A firm has the marginal revenue function given by MR = \(\frac { a }{ { (x+b) }^{ 2 } } \) - c where x is the output and a, b, c are constants. Show that the demand function is given by \(x=\frac { a }{ b(p+c) } -b\).

The marginal cost C'(x) and marginal revenue R'(x) are given by C'(x) = 50 + \(\frac{x}{50}\) and R'(x) = 60. The fixed cost is Rs. 200. Determine the maximum profit

The marginal cost and marginal revenue with respect to commodity of a firm are given by C'(x) = 8 + 6x and R'(x)= 24. Find the total Profit given that the total cost at zero output is zero.

The marginal revenue function (in thousand of rupees ) of a commodity is 10 + e^−0.05x Where x is the number of units sold. Find the total revenue from the sale of 100 units (e⁻⁵ = 0.0067)

The price of a machine is Rs. 5,00,000 with an estimated life of 12 years. The estimated salvage value is Rs. 30,000. The machine can be rented at Rs. 72,000 per year. The present value of the rental payment is calculated at 9% interest rate. Find out whether it is advisable to rent the machine.(e^−1.08 = 0.3396).

Elasticity of a function \(\frac{Ey}{Ex}\) is given by \(\frac{Ey}{Ex}\) = \(​​\frac { -7x }{ (1-2x)(2+3x) } \). Find the function when x = 2, y = \(\frac{3}{8}\)

The elasticity of demand with respect to price for a commodity is given by \(\frac{(4-x)}{x}\), where p is the price when demand is x. Find the demand function when price is 4 and the demand is 2. Also find the revenue function.

When the Elasticity function is \(\frac { x }{ x-2 } \). Find the function when x = 6 and y = 16.

The elasticity of demand with respect to price p for a commodity is \(\eta _{ d }=\frac { p+2{ p }^{ 2 } }{ 100-p-{ p }^{ 2 } } \).Find demand function where price is Rs. 5 and the demand is 70.

A firm’s marginal revenue function is MR = 20e^-x/10 \(\left( 1-\frac { x }{ 10 } \right) \). Find the corresponding demand function.

The marginal cost of production of a firm is given by C'(x) = 5 + 0.13x, the marginal revenue is given by R'(x) = 18 and the fixed cost is Rs. 120. Find the profit function.

If the marginal cost (MC) of a production of the company is directly proportional to the number of units (x) produced, then find the total cost function, when the fixed cost is Rs. 5,000 and the cost of producing 50 units is Rs. 5,625.

The demand and supply function of a commodity are p_d = 18− 2x − x² and p_s = 2x − 3 . Find the consumer’s surplus and producer’s surplus at equilibrium price.

Under perfect competition for a commodity the demand and supply laws are P_d  =  \(\frac { 8 }{ x+1 } -2\) and P_s = \(\frac { x-3 }{ 2 } \) respectively. Find the consumer’s and producer’s surplus.

The demand equation for a products is x = \(\sqrt { 100-p } \) and the supply equation is x = \(\frac{p}{2}\) -10. Determine the consumer’s surplus and producer’s surplus, under market equilibrium.

Find the consumer’s surplus and producer’s surplus for the demand function p_d = 25 − 3x and supply function p_s = 5 + 2x.

The marginal cost of production of a firm is given by C'(x) = 20 + \(\frac { x }{ 20 } \) the marginal revenue is given by R'(x) = 30 and the fixed cost is Rs. 100. Find the profit function

The demand equation for a product is p_d = 20 − 5x and the supply equation is p_s = 4x + 8. Determine the consumer’s surplus and producer’s surplus under market equilibrium.

A company requires f(x) number of hours to produce 500 units. It is represented by f (x) = 1800x^−0.4. Find out the number of hours required to produce additional 400 units. [(900)^0.6 = 59.22, (500)^0.6 = 41.63]

The price elasticity of demand for a commodity is \(\frac { p }{ { x }^{ 3 } } \). Find the demand function if the quantity of demand is 3, when the price is Rs. 2