(1) The integral is straightforward; <em>x</em> ranges between two constants, and <em>y</em> ranges between two functions of <em>x</em> that don't intersect.
(2) First find where the two curves intersect:
<em>y</em> ² - 4 = -3<em>y</em>
<em>y</em> ² + 3<em>y</em> - 4 = 0
(<em>y</em> + 4) (<em>y</em> - 1) = 0
<em>y</em> = -4, <em>y</em> = 1 → <em>x</em> = 12, <em>x</em> = -3
That is, they intersect at the points (-3, 1) and (12, -4). Since <em>x</em> ranges between two explicit functions of <em>y</em>, you can capture the area with one integral if you integrate with respect to <em>x</em> first:
(3) No special tricks here, <em>x</em> is again bounded between two constants and <em>y</em> between two explicit functions of <em>x</em>.
Life, liberty, pursuit of happiness. The reason these are at the very beginning of the document is because without these life has no meaning. These also use the dominion effect. without life there is no liberty, without liberty there can be no true happiness.
Our understanding of the universe has changed a lot over time!
Ptolemy
He improved on Aristotle's theory which was that the earth was the center of the universe also known as geocentric.
Nicolaus Copernicus
Copernicus had a theory that the sun was the center of our solar system. This is also known as heliocentric.
Tycho Brahe
His work supported the idea that the Earth orbited the sun.
Brahe hired Kepler, another astronomer to work with him.
How has our understanding of the universe changed over time?
Aristotle
Johannes Kepler
Galileo
Isaac Newton
Edmund Halley
Edwin Hubble
By Isamilla & Nelleke
The Hubble space telescope was named after Edwin Hubble.
He had three laws of motion.
He hypothosized that the nebulae were their own galaxies.
<span>He imagined the earth at the </span>
center of the universe with the sun, the moon, the stars and the planets orbiting it in perfect circles.
Newton put Kepler and Galileo's ideas together and discovered that the reason that things fall on the ground is the same reason that planets orbit around the sun.
He made a telescope that could magnify
up to 30 times.
<span>He discovered that Copernicus' theory </span>
was true. "We're not the center of the universe."
He found out that the planets orbit the
sun in ellipses (oval shapes).
He explained how the tides were made by the moon.
He figured out that the speed of the planets depend on the sun.
Halley discovered a comet. It was discovered in 1705. It will come around again in 2061
The comet Halley comes every 75 - 76 years
384–322 BCE
90 AD – 168 AD
1473 – 1543
1546 – 1601
1571 – 1630
1564 – 1642
1642 – 1727
1656 – 1742
1889 – 1953
Celestial Sphere
<span>It's an invisible sphere around the earth
</span>
Odd question...but, uh...if it helps...grab the stick that's in your eye and pull yourself to safety?
Answer:
Yes.
Explanation:
Yes, there is a great development occur in the Philippines. The Philippine's economy has made progress in the growth which can be seen in the lowering occurs in the poverty rate and unemployment rate in the country. There is a significant decrease in the poverty rate in the country due to investment of foreign companies in the Philippines in the sector of tourism and other industries which decreases the unemployment rate in the country that leads to decrease poverty rate also.
