(8) Determine the area to the curve y = x2 + 7x + 1 and x = 1 and x = 2. (9) Determine the bounded area of…

Question

(8) Determine the area to the curve y = x2 + 7x + 1 and x = 1 and x = 2. (9) Determine the bounded area of the curve y = x2 - 3x + 4.

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8) Step 1: Set up the definite integral for the area under the curve y = x² + 7 from x=1 to x=2. A = ∫₁^2 (x² + 7) dx Step 2: Integrate the function term by term. ∫ (x² + 7) dx = (x²⁺¹)/(2+1) + 7x = (x³)/(3) + 7x Step 3: Evaluate the definite integral using the Fundamental Theorem of Calculus. A = [ (x³)/(3) + 7x ]₁^2 = ( (2³)/(3) + 7(2) ) − ( (1³)/(3) + 7(1) ) Step 4: Calculate the values. A = ( 8/3 + 14 ) − ( 1/3 + 7 ); A = ( 8/3 + 42/3 ) − ( 1/3 + 21/3 ); A = 50/3 − 22/3 = 28/3 The area is 28/3 square units. 9) Step 1: Set up the definite integral for the area under the curve y = x² − 3x + 4 from x=0 to x=4. A = ∫₀^4 (x² − 3x + 4) dx Step 2: Integrate the function term by term. ∫ (x² − 3x + 4) dx = (x²⁺¹)/(2+1) − 3(x¹⁺¹)/(1+1) + 4x = (x³)/(3) − (3x²)/(2) + 4x Step 3: Evaluate the definite integral using the Fundamental Theorem of Calculus. A = [ (x³)/(3) − (3x²)/(2) + 4x ]₀^4 = ( (4³)/(3) − (3(4²))/(2) + 4(4) ) − ( (0³)/(3) − (3(0²))/(2) + 4(0) ) Step 4: Calculate the values. A = ( 64/3 − (3(16))/(2) + 16 ) − (0); A = 64/3 − 48/2 + 16; A = 64/3 − 24 + 16; A = 64/3 − 8 Step 5: Combine the terms. A = 64/3 − 24/3 = 40/3 The area is 40/3 square units.

(8) Determine the area to the curve y = x2 + 7x + 1 and x = 1 and x = 2. (9) Determine the bounded area of the curve y = x2 − 3x + 4.

Step-by-step solution

Set up the definite integral for the area under the curve $y = x^2 + 7$ from $x=1$ to $x=2$ .

Integrate the function term by term.

Evaluate the definite integral using the Fundamental Theorem of Calculus.

Calculate the values.

Set up the definite integral for the area under the curve $y = x^2 - 3x + 4$ from $x=0$ to $x=4$ .

Integrate the function term by term.

Evaluate the definite integral using the Fundamental Theorem of Calculus.

Calculate the values.

Combine the terms.

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More Mathematics Questions

(8) Determine the area to the curve y = x2 + 7x + 1 and x = 1 and x = 2. (9) Determine the bounded area of the curve y = x2 − 3x + 4.

Step-by-step solution

Set up the definite integral for the area under the curve y=x2+7y = x^2 + 7y=x2+7 from x=1x=1x=1 to x=2x=2x=2.

Integrate the function term by term.

Evaluate the definite integral using the Fundamental Theorem of Calculus.

Calculate the values.

Set up the definite integral for the area under the curve y=x2−3x+4y = x^2 - 3x + 4y=x2−3x+4 from x=0x=0x=0 to x=4x=4x=4.

Integrate the function term by term.

Evaluate the definite integral using the Fundamental Theorem of Calculus.

Calculate the values.

Combine the terms.

Still stuck on this one?

More Mathematics Questions

Set up the definite integral for the area under the curve $y = x^2 + 7$ from $x=1$ to $x=2$ .

Set up the definite integral for the area under the curve $y = x^2 - 3x + 4$ from $x=0$ to $x=4$ .