Logo

AskSia

Plus

Calculate the area under the curve y=x3y = x^3 from x=1x = 1 to x=4x = 4.
May 5, 2024
Calculate the area under the curve y=x3y = x^3 from x=1x = 1 to x=4x = 4.
Generated Graph
Solution by Steps
step 1
To find the area under the curve y=x3y = x^3 from x=1x = 1 to x=4x = 4, we need to evaluate the definite integral ∫14x3 dx\int_{1}^{4} x^3 \, dx
step 2
The antiderivative of x3x^3 is x44\frac{x^4}{4}
step 3
We evaluate the antiderivative at the upper and lower limits of the integral: [x44]14\left[\frac{x^4}{4}\right]_{1}^{4}
step 4
Plugging in the upper limit, we get 444=2564=64\frac{4^4}{4} = \frac{256}{4} = 64
step 5
Plugging in the lower limit, we get 144=14\frac{1^4}{4} = \frac{1}{4}
step 6
Subtracting the lower limit evaluation from the upper limit evaluation gives us the area: 64−14=2564−14=255464 - \frac{1}{4} = \frac{256}{4} - \frac{1}{4} = \frac{255}{4}
Answer
The area under the curve y=x3y = x^3 from x=1x = 1 to x=4x = 4 is 2554\frac{255}{4} or 63.7563.75.
Key Concept
Definite Integral as Area Under the Curve
Explanation
The definite integral of a function from a to b gives the area under the curve of the function between the vertical lines x=ax = a and x=bx = b. In this case, the function is y=x3y = x^3, and the limits are from x=1x = 1 to x=4x = 4.
Generated Graph
Solution by Steps
step 1
To find the perimeter of the shaded sector, we add the radius of the circle to the arc length twice, since the sector includes two radii and the arc
step 2
The radius is given as 4 cm, and the arc length is given as 10 cm
step 3
The perimeter of the sector is therefore 2×4+10=8+10=182 \times 4 + 10 = 8 + 10 = 18 cm
[question 1] Answer
The perimeter of the shaded sector is 18 cm.
Key Concept
Perimeter of a Sector
Explanation
The perimeter of a sector is the sum of the lengths of the two radii and the arc length.
Solution by Steps
step 1
To find the central angle θ\theta in radians, we use the formula for the arc length of a circle, s=rθs = r\theta, where ss is the arc length, rr is the radius, and θ\theta is the central angle in radians
step 2
We are given that the arc length s=10s = 10 cm and the radius r=4r = 4 cm
step 3
Solving for θ\theta, we have θ=sr=104=52\theta = \frac{s}{r} = \frac{10}{4} = \frac{5}{2} radians
[question 2] Answer
The central angle θ\theta is 52\frac{5}{2} radians.
Key Concept
Central Angle of a Sector
Explanation
The central angle of a sector, measured in radians, is found by dividing the arc length by the radius.
Solution by Steps
step 1
To find the area of the shaded sector, we use the formula for the area of a sector of a circle, A=12r2θA = \frac{1}{2}r^2\theta, where AA is the area, rr is the radius, and θ\theta is the central angle in radians
step 2
We are given that the radius r=4r = 4 cm and the central angle θ=52\theta = \frac{5}{2} radians
step 3
Substituting the given values into the formula, we get A=12×42×52=12×16×52=20A = \frac{1}{2} \times 4^2 \times \frac{5}{2} = \frac{1}{2} \times 16 \times \frac{5}{2} = 20 cm2^2
[question 3] Answer
The area of the shaded sector is 20 cm2^2.
Key Concept
Area of a Sector
Explanation
The area of a sector is calculated by multiplying half the square of the radius by the central angle in radians.
© 2023 AskSia.AI all rights reserved